Laurentiu Hetel
email: laurentiu.hetel@ec-lille.fr

Jamal Daafouz
email: jamal.daafouz@ensem.inpl-nancy.fr

Sophie Tarbouriech

Christophe Prieur
email: christophe.prieur@gipsa-lab.grenoble-inp.fr

Reset control systems: stabilization by nearly-periodic reset

Keywords: Impulsive systems, Reset laws, Stability Analysis, Stabilization, Uncertainty, !"#$%&'#(%)*+,

In this paper the class of linear impulsive systems is considered. These systems are those with a continuous linear dynamics for all time, except at a sequence of instants. When such a discrete time occurs, the state undergoes a jump, or more precisely follows a discrete linear dynamics. The sequence of time instants, when a discrete dynamics occurs, is nearlyperiodic only, i.e. it is distant from a periodic sequence to an uncertain distance. This paper succeeds to state tractable conditions to analyze the stability, and to design reset matrices such that the hybrid system is globally asymptotically stable to the origin. The approach is based on a polytopic embedding of the uncertain dynamics. An example illustrates the main stability result.

INTRODUCTION

Hybrid systems are dynamic systems subject to both continuous-time and discrete-time dynamics. The importance of hybrid systems in control systems analysis and design have been growing in the last decades, due mainly to their presence in practical systems and to overcome performance limitations of more classical controllers, i.e., regular linear or nonlinear controllers (see, e.g., [START_REF] Clegg | A non-linear integrator for servomechanisms[END_REF], [START_REF] Prieur | Hybrid feedback control and robust stabilization of nonlinear systems[END_REF], [START_REF] Zaccarian | First order reset elements and the Clegg integrator revisited[END_REF], [START_REF] Nesic | Stability properties of reset systems[END_REF]). A particular class of hybrid systems is represented by reset systems, see [START_REF] Horowitz | Non-linear design for cost of feedback reduction in systems with large parameter uncertainty[END_REF], [START_REF] Beker | Plant with an integrator: an example of reset control overcoming limitations of linear feedback[END_REF], [START_REF] Prieur | Guaranteed stability for nonlinear systems by means of a hybrid loop[END_REF], [START_REF] Barreiro | Delay-dependent stability of reset systems[END_REF]. An interesting sub-class of hybrid systems is the systems with finite state jump which are linear continuous-time systems for which the state undergoes finite jump discontinuities at some discrete instants of time. Such systems can be regarded as a special case of reset systems in the sense that the reset rule is done through a time condition instead of a state condition. These systems can be named impulsive dynamical linear systems [START_REF] Haddad | Impulsive and hybrid dynamical systems[END_REF]. See also for example Haddad et al. [2001a], 1 The research leading to these results has received funding from the ANR project ArHyCo, ARPEGE, contract number ANR-2008 SEGI 004 01-30011459 and from the European Community's Seventh Framework Programme (FP7/2007(FP7/ -2013) ) under grant agreement n 257462: HYCON2 Network of Excellence "Highly-Complex and Networked Control Systems". Haddad et al. [2001b], [START_REF] Li | Analysis and design of impulsive control systems[END_REF] and [START_REF] Hespanha | Lyapunov conditions for input-to-state stability of impulsive systems[END_REF]. Lyapunov theory framework provides the main tool to test the stability of reset of impulsive systems by employing an adequate Lyapunov function (or a family of Lyapunov functions) [START_REF] Liberzon | Switching in Systems and Control[END_REF]. The results developed in the current paper are based on the use of adequate parameter-dependent quadratic Lyapunov function [START_REF] Lee | Uniform stabilization of discrete-time switched and markovian jump linear systems[END_REF], [START_REF] Daafouz | Parameter dependent Lyapunov functions for discrete time systems with time varying parametric uncertainties[END_REF]).

The paper deals with both the stability analysis and stabilization problems, for linear systems controlled by a reset compensator. The reset rule, differently from [START_REF] Prieur | Guaranteed stability for nonlinear systems by means of a hybrid loop[END_REF], is posed in function of the time. The interval between two reset instants is supposed to be uncertain. In this setup, this interval is defined as the sum of a nominal reset period and an uncertain time-varying term bounded in a given interval. Thus, in order to avoid Zenon phenomena, we assume that the nominal reset period is different from zero. The main conditions in both problems is exhibited in terms of a parametric set of linear matrix inequalities (LMI). Tractable numerical solutions are proposed for these sets of parametric LMI to be expressed as a finite number of conditions using convex embeddings.

The paper is organized as follows. Section 2 describes the class of systems under consideration and states the problems of stability analysis and reset law design. In Section 3, the main results to with both problems. Section 4 presents two ways to solve the conditions developed in Section 3. An numerical example is detailed in Section 5 to point out the potentiality but also the difficulty of the proposed approach. Finally some concluding remarks and forthcoming issues end the paper.

Notation. For a matrix M we denote by M the induced matrix norm. By M 0 or M ≺ 0 we mean that the symmetric matrix M is positive or negative definite respectively. We denote the transpose of a matrix M by M T . By I m we denote the m × m identity matrix. By 0 we denote the null matrix of the appropriate dimension. By λ max (M ) we denote the maximum eigenvalue of a square symmetric matrix. For a given set S, co(S) denotes the convex hull of S.

SYSTEM DESCRIPTION

Consider the following linear time-invariant system:

ẋp (t) = Ax p (t) + Bu(t), ∀t ∈ R + , x p (0) = x 0 p ∈ R np (1) y(t) = Cx p (t),
where x p ∈ R np , u ∈ R nu and y ∈ R ny represent the system state, input and output, respectively. The matrices A, B and C are constant matrices of appropriate dimension. Associated to the system (1) we consider a reset controller:

η(t) = A η η(t) + B η y(t), ∀t ∈ R + -T , η(0) = η 0 ∈ R nη u(t) = C η η(t) + D η y(t), (2)
η(t) = R η η(t -) + R y y(t -), ∀t ∈ T ,
where the matrices A η , B η , C η , D η , R y , R η are constant matrices of appropriate dimmension. η ∈ R nη is the controller state and t -= lim τ →t,τ <t t. The set

T = t k : t k ∈ R + , t k < t k+1 , ∀k ∈ N, lim k→∞ t k = ∞ (3)
represents the set of reset times. We denote by τ k the interval between two reset instants,

τ k := t k+1 -t k , ∀ k ∈ N.
(4) We assume that the reset interval has the form

τ k = τ nom + δτ k , (5)
where τ nom represents a nominal reset period and δτ k the uncertain time-varying term bounded in a given interval,

δτ k ∈ [0, δτ max ] . (6
) In order to avoid Zenon phenomena we assume that τ nom = 0. The problems under study are formulated as follows:

Problem 1. Assume that the matrices A, B, C and A η , B η , C η , D η , R η and R y are given and constant. Provide LMI methods for checking the stability of the reset system (1), (2) with a reset interval (5).

Problem 2. Assume that the matrices A, B, C and A η , B η , C η , D η are given and constant. Design reset matrices R η and R y to guarantee stability of reset system (1), (2) with a reset interval (5).

RESET STABILIZATION

We consider the following generic linear reset systems:

ẋ(t) = A c x(t), ∀t ∈ R + -T , (7)
x(t) = A r x(t -), ∀t ∈ T . (8)
Note that the system (1) with the control law (2) can be expressed in this form using the notation

x = (x T p η T ) T ∈ R n , n = n p + n η A c = A + BD η C BC η B η C A η (9)
and

A r = I np 0 R y C R η . ( 10
)
The following theorem proposes stability conditions for closed-loop system (7) with a reset law (8).

Theorem 1. Consider system ( 7),( 8), with T defined in (3) such that the reset interval (4) satisfies the assumptions ( 5) and ( 6). Assume that the matrix A r is given. If there exist symmetric positive definite matrices P (δτ ), δτ ∈ [0, δτ max ], and a matrix G of appropriate dimensions such that the following set of linear matrix inequalities

P (δτ a ) e Ac(τnom+δτa T A T r G T GA r e Ac(τnom+δτa) G + G T -P (δτ b ) 0, (11)
is satisfied for all δτ a , δτ b ∈ [0, δτ max ], then there exists a reset law (8) for which the equilibrium point x = 0 of system ( 7) is asymptotically stable.

Proof. The proof is based on the evaluation of the system behavior at the sampling instants t k . The solution of the system ( 7) is described by the equation

x(t) = e (t-t k )Ac x(t k ), ∀t ∈ [t k , t k+1 ) .
(12) Note that since the continuous-time dynamic in between two resets instances is linear, then the evolution of the norm of the system state is upper-bounded for all t ∈ [t k , t k+1 ), i.e. there exist a scalar =

λ max A c + A T c 2 s.t. x(t) ≤ e (t-t k ) x(t k ) for all t ∈ [t k , t k+1
). This implies that for analyzing the asymptotic stability of the system it suffice to analyze the behavior of the system for the values t = t k , k ∈ N. Note that at t = t k+1 the system state is given by:

x(t k+1 ) = A r e (t k+1 -t k )Ac x(t k ) = A r e τ k Ac x(t k ) = A r e (τnom+δτ k )Ac x(t k )
( 1 3 ) The previous equation represents a discrete-time linear system with time-varying parameters δτ k that appear in an exponential manner. Following generic stability results for linear system with time-varying parameters [START_REF] Lee | Uniform stabilization of discrete-time switched and markovian jump linear systems[END_REF], [START_REF] Daafouz | Parameter dependent Lyapunov functions for discrete time systems with time varying parametric uncertainties[END_REF]), we consider the following class of Lyapunov functions:

V (x(t k ), δτ k ) = x T (t k )P (δτ k )x(t k ).
(14) The equilibrium point x = 0 is stable if the function is strictly decreasing for all x(t k ) = 0 and all variations of the parameter δτ k . This condition can be expressed as

V (x(t k+1 ), δτ k+1 ) < V (x(t k ), δτ k ) ( 1 5 )
!"#$%&'#(%)*+,*-.-/01,2-32456/-78+,$73,309-0:,785;+ <30=3-8/,>4?.-//01,/7,@A/B,%$)!,C7351,!78D30>>+,E020-901,"2/7?03,@FG,HI@I+ which is the same as

V A r e (τnom+δτ k )Ac x(t k ), δτ k+1 < V (x(t k ), δτ k ) (16)
for all δτ k , δτ k+1 in [0, δτ max ] and for all x(t k ) = 0. Then the stability is ensured if

max δτ b ∈[0,δτmax] V A r e (τnom+δτa)Ac x, δτ b < V (x, δτ a ) (17)
holds for all δτ a ∈ [0, δτ max ], x = 0.

Assume that there exist matrices P (δτ ), δτ ∈ [0, δτ max ], and a matrix G of appropriate dimensions such that the condition ( 11) is satisfied ∀ δτ a , δτ b ∈ [0, δτ max ]. Using similar arguments as in [START_REF] De Oliveira | A new discrete-time robust stability condition[END_REF]), one can see that multiplying the inequality (11) by

T := [I n -(A r e Ac(τnom+δτa)
) T ] on the left and by its transpose on the right, the following inequality holds true:

e Ac(τnom+δτa T A T r P (δτ b )A r e Ac(τnom+δτa) -P (δτ a ) ≺ 0, (18) ∀ δτ a , δτ b ∈ [0, δτ max ] which guarantees that the condition ( 17) is satisfied. 2

The following theorem proposes conditions for the design of a reset matrix A r , as defined in (8), that stabilizes the system (7). Theorem 2. Consider system (7) with T defined in (3) such that the reset interval (4) satisfies the assumptions ( 5) and ( 6). If there exist symmetric positive definite matrices P (δτ a ), δτ a ∈ [0, δτ max ], and matrices G, W of appropriate dimensions such that the following set of linear matrix inequalities

P (δτ a ) e Ac(τnom+δτa T W T W e Ac(τnom+δτa) G + G T -P (δτ b ) 0, ( 19
)
is satisfied for all δτ a , δτ b ∈ [0, δτ max ], then there exists a reset law (8) for which the equilibrium point x = 0 of system ( 7) is asymptotically stable. The reset law is given by ( 8) with A r = G -1 W.

Proof. Assume that there exist matrices P (δτ ), δτ ∈ [0, δτ max ], and matrices G, W of appropriate dimension such that the condition ( 19) is satisfied. Using the change of variables W = GA r the set of inequalities (19) leads to

P (δτ a ) e Ac(τnom+δτa T A T r G T GA r e Ac(τnom+δτa) G + G T -P (δτ b ) 0, (20) ∀ δτ a , δτ b ∈ [0, δτ max ].
Using Theorem 1 guarantees the asymptotic stability. 2

In the following corollary it is shown how the approach in Theorem 2 can be used in order to develop reset law for the particular case of system (1) with the reset law (2). Note that in this case the reset matrix A r has a particular structure, as described in (10).

Corollary 1. Consider system (1), the reset control (2) and the closed-loop matrices ( 9), (10). Moreover, consider that the set T defined in (3) with the reset interval (4) satisfies the assumptions ( 5) and ( 6). If there exist symmetric positive definite matrices P (δτ a ), δτ a ∈ [0, δτ max ], and matrices Ḡ, W η , W y of appropriate dimensions such that the set of linear matrix inequalities ( 19) is satisfied with

G = I np 0 0 Ḡ (21)
and

W = I np 0 W y C W η (22)
for all δτ a , δτ b ∈ [0, δτ max ], then there exist a reset law (8) with a structured reset matrix (10) for which the equilibrium point x = 0 of system (1) is globally uniformly exponentially stable. The reset law is given by ( 8) and ( 10) with R y = Ḡ-1 W y and R η = Ḡ-1 W η .

Proof. Assume that the exist symmetric positive definite matrices P (δτ a ), δτ a ∈ [0, δτ max ], and matrices G and W as in ( 21), ( 22) such that conditions (19) are satisfied.

According to Theorem 2, there exists a stabilizing reset matrix A r given by

A r = G -1 W (23) = I np 0 0 Ḡ-1 I np 0 W y C W η (24) = I np 0 Ḡ-1 W y C Ḡ-1 W η , (25)
which is of the form (10

) with R y = Ḡ-1 W y and R η = Ḡ-1 W η . 2
The application of the previous corollary to the ideal case in which no uncertainty concerns the reset interval (δτ max = 0) is given as follows:

Corollary 2. Consider system (1), the reset control (2) and the closed-loop matrices ( 9), (10). Moreover, consider that the set T defined in (3) with a nominal reset interval τ nom . If there exist a symmetric matrix P and matrices Ḡ, W η , W y of appropriate dimension such that the set of linear matrix inequalities ( 19) is satisfied for P (δτ a ) = P, δτ a ∈ [0, δτ max ], with G and W as defined in ( 21) and ( 22), respectively, then there exists a reset law (8) with the structured reset matrix (10) for which the equilibrium point x = 0 of system (1) is asymptotically stable. The reset law is given by ( 8) and ( 10) with R y = Ḡ-1 W y and R η = Ḡ-1 W η .

Remark. Corollary 2 presents stabilization condition for the case of nominal reset interval. The given conditions represent a finite set of linear matrix inequalities which can be checked using classical convex optimization tools. Note that the conditions ( 11), ( 19) lead to a parametric set of linear matrix inequalities, since they depend on the different values of δτ a , δτ b in the interval [0, δτ max ].

Tractable numerical condition for solving this parametric set of conditions are presented in the following section.

NUMERICAL EVALUATION

In this section it is shown how the parametric set of LMIs presented previously, namely ( 11), ( 19), can be expressed as a finite number of conditions using convex embeddings.

Polytopic sets

In order to obtain a finite number of LMIs from ( 11) and ( 19), we have to deal with the exponential uncertainty !"#$%&'#(%)*+,*-.-/01,2-32456/-78+,$73,309-0:,785;+ <30=3-8/,>4?.-//01,/7,@A/B,%$)!,C7351,!78D30>>+,E020-901,"2/7?03,@FG,HI@I+ e Acδτ k [START_REF] Hetel | LMI control design for a class of exponential uncertain systems with application to network controlled switched systems[END_REF]) that appears in the conditions. Note that the uncertain matrices e Acδτ k are continuous with respect to δτ k and that δτ k belongs to a bounded set.

Then, for δτ a ∈ [0, δτ max ], the matrix e Acδτa describes a compact subset of R n×n : E = X ∈ R n×n : X = e Acδτa , δτ a ∈ [0, δτ max ] (26) The basic idea is to embed the set of matrices E into a polytopic set Z, i.e. to find a set of N matrices Z i such that E ⊂ Z = co {Z 1 , Z 2 , . . . , Z N } .

(27) This implies that for all δτ a ∈ [0, δτ max ] there exist a set of scalars µ i (δτ a ) ∈ [0, 1], i = 1, . . . , N, such that

e Acδτa = N i=1 µ i (δτ a )Z i , N i=1 µ i (δτ a ) = 1. ( 28
)
In order to provide some insight about how a polytopic embedding can be approximated, we recall briefly some of the methods proposed in [START_REF] Hetel | LMI control design for a class of exponential uncertain systems with application to network controlled switched systems[END_REF]).

Consider an h-order Taylor series expansion of the matrix exponential e Acδτ :

e δτaAc ≈ h l=0 A c l l! δτ l a . ( 29
)
Then we may construct a convex polytope by considering the terms δτ l , l = 1, 2, . . . , h, as independent parameters. The h-order Taylor approximation can be embedded in a matrix hypercube with 2 h vertices. However, one can exploit the relation between the different parameters to construct a convex polytope inside the hypercube. This approached is mathematically formalized in Lemma 4 (given in the Appendix). Applying Lemma 4 to the polynomial form (29) for δτ a ∈ [0, δτ max ], leads to a polytopic approximation ( 27) with N = h + 1 vertices given by:

Z 1 = I n , Z 2 = δτ max A c + I n , Z 3 = δτ 2 max A c 2 2! + δτ max A c + I n , . . . Z h+1 = δτ h max A c h h! + δτ h-1 max A c h-1 (h -1)! + . . . . . . . + δτ 2 max A c 2 2! + δτ max A c + I n .
In the following subsection we illustrate the use of such polytopic embedding methods for reset matrices design.

Stabilization based on polytopic sets

As follows we show how the parametric set of linear matrix inequalities, such as ( 11), ( 19), can be reduced to a finite number of linear matrix inequalities. The approach is illustrated for the condition (19). The conditions proposed in Theorem 1 and Corolarry 1, can be treated in a similar manner. The reset matrix design procedure based on polytopic set ( 27) is formulated in the following Theorem. Theorem 3. Consider system (7) with T defined in (3) such that the reset interval (4) satisfies the assumptions (5) and ( 6). Moreover, consider the polytopic set ( 27) with N vertices. If there exists symmetric positive definite matrices P i , i = 1, . . . , N, and matrices G, W of appropriate dimensions such that the following set of linear matrix inequalities

P i Z T i e Acτnom T W T W e Acτnom Z i G + G T -P j 0, (30)
is satisfied for all i, j = 1, . . . , N, then there exists a reset law (8) for which the equilibrium point x = 0 of system ( 7) is asymptotically stable. The reset law is given by ( 8) with A r = G -1 W.

Proof. Assume that there exist a set of matrices P i , i = 1, . . . , N and matrices G and R such that the set of inequalities (30) hold true. Then the condition

N i=1 µ i P i N i=1 µ i Z T i e Acτnom W T N i=1 µ i W e Acτnom Z i G + G T - N i=1 µ j P j 0, (31)
is satisfied for any set of scalars µ i , µ j ∈ [0, 1], i, j = 1, . . . , N, such that N i=1 µ i = N i=1 µ j = 1. Note that e Acδτa ∈ Z for all δτ a ∈ [0, δτ max ]. Then the previous condition implies the existences of matrices P (δτ a ) such that the condition (19) holds with !"#$%&'#(%)*+,*-.-/01,2-32456/-78+,$73,309-0:,785;+ <30=3-8/,>4?.-//01,/7,@A/B,%$)!,C7351,!78D30>>+,E020-901,"2/7?03,@FG,HI@I+ For this system the matrix A r e τnom is Schur for τ nom ∈ [0, 0.58]. This implies that the system is stable if the reset occurs periodically, with constant reset interval τ nom in [0, 0.58] and δτ k = 0. However, variation of the reset interval may induce instability. While the system is stable for τ 1 nom = 0.28 and τ 2 nom = 0.54 (see Fig. 1 and Fig. 2, respectively), the matrix A r e τ 1 nom A r e τ 2 nom has eigenvalues outside the unit circle. Therefore, resetting the system with the pattern τ 1 nom → τ 2 nom → τ 1 nom → τ 2 nom . . ., leads to an unstable behavior. This phenomena is illustrated in Fig. 3.

P (δτ a ) = N i=1 µ i (δτ a )P i , (32)
P (δτ b ) = N j=1 µ j (δτ b )P j , (33)
Consider now that τ nom = 0.1 and that δτ max = 0.2. In order to illustrate graphically the construction of the polytopic embedding Z in (27), consider the Jordan normal form of A c : . . . , 3, given in (35). For this particular case, the exponential uncertainty e δτ k J has the form

J = λ 1 0 0 0 λ 2 0 0 0 λ 3 . (36) with λ i = 1,
e δτ k J = T φ 1 (δτ k ) 0 0 0 φ 2 (δτ k ) 0 0 0 1 T -1 (37)
with two uncertain scalar parameters φ i (δτ k ) = e δτ k λi , i = 1, 2, and δτ k ∈ [0, 0.2]. In Fig. 4, we illustrate the curve described by the exponential uncertainty in the φ 1 (δτ k )φ 2 (δτ k ) plane as well as the obtained polytopic embedding based on a 8th order Taylor series expansion (in dark gray). Note that the polytopic embedding procedure can be refined by dividing the interval [0, δτ max ] in several subintervals, and applying the Taylor method locally. Then, the extreme point of the global polytope can be determined among the vertex of each local embedding using classical convex hull algorithms. An illustration based on 5 subinterval is given in Fig. 4 (light gray). Using such an embedding and adapting Theorem 1 similarly to Theorem 3, we can show that the system is robustly stable for τ nom = 0.1 with variations of the reset interval of a maximum amplitude δτ max = 0.2. An example of system evolution according to a random variation of the reset interval is given in Fig. 5.

CONCLUSION

Focusing on linear impulsive systems, we stated tractable conditions to analyze the stability. Also some sufficient conditions are derived to compute reset matrices such that the hybrid system is global asymptotically stable to the origin. It is assumed that the jump instants are not periodic but only nearly-periodic, i.e. with an uncertain distance to a periodic sequence.

This paper lets many questions open. The first one may be the generalization of this work to nonlinear systems. Lyapunov techniques may be applied in this context (see e.g. [START_REF] Goebel | Smooth patchy control Lyapunov functions[END_REF]). Also the performance issues may be considered either for linear impulsive systems or for nonlinear ones. One criterion to be optimized may be the rejection of perturbations and the gain perturba-!"#$%&'#(%)*+,*-.-/01,2-32456/-78+,$73,309-0:,785;+ <30=3-8/,>4?.-//01,/7,@A/B,%$)!,C7351,!78D30>>+,E020-901,"2/7?03,@FG,HI@I+

Fig. 1 .

 1 Fig. 1. Illustration of system evolution with a periodic reset interval τ 1 nom = 0.28.

  where µ i (δτ a ), µ j (δτ b ) ∈ [0, 1], i, j = 1, . . . , N, represent the barycentric coordinates of e Acδτa and e Acδτ b in the polytope Zmatrix A c has the eigenvalues λ 1 = -1.44, λ 2 = 3.44, λ 3 = 0. (35)

Fig. 2 .Fig. 3 .

 23 Fig. 2. Illustration of system evolution with a periodic reset interval τ 2 nom = 0.54.

  Representation of exponential uncertainty and of construction of convex polytopes in the φ 1 (δτa)φ 2 (δτ b ) plane, for τnom = 0.1, δτa ∈ [0, 0.2]. The polytopic embedding in dark gray is obtained based on a 8th order Taylor approximation. In light gray we represent a polytopic embedding obtained based on a 8th order Taylor approximation applied on 5 subintervals.

	1
	0.95
	0.9
	0.85
	0.8
	0.75
	0.7
	0.65
	0.6
	0.55	1	1.2	1.4	1.6	1.8	2	2.2	2.4	2.6	2.8	3
	Fig. 4.

tions/output to be made as low as possible. The connection with the Input-to-State Stability (ISS) of impulsive systems as in [START_REF] Hespanha | Lyapunov conditions for input-to-state stability of impulsive systems[END_REF] may be fruitful.

APPENDIX

Lemma 4. [START_REF] Hetel | LMI control design for a class of exponential uncertain systems with application to network controlled switched systems[END_REF]) Consider a polynomial matrix

where the parameter ρ ∈ R and L i ∈ R n×n , i = 1, . . . , p. For each upper bound ρ ≥ 0 on ρ there exist matrices U i ∈ R n×n , i = 1, . . . , p + 1, s.t. the following property holds: For all ρ ∈ [0, ρ] there exist parameters µ i (ρ), i = 1, . . . , p+ 1, with p+1 i=1 µ i (ρ) = 1, and µ i (ρ) ≥ 0, i = 1, . . . , p + 1, s.t.

In particular, U i , i = 1, . . . , p, can be chosen as

The relations between the parameters ρ and µ are given by:

µ p+1 (ρ) = 1 -ρ p ρ p .


Christophe Prieur
email: christophe.prieur@gipsa-lab.grenoble-inp.fr

Sophie Tarbouriech
email: sophie.tarbouriech@laas.fr

Luca Zaccarian

Improving the performance of linear systems by adding a hybrid loop

Keywords: hybrid systems, performance, reset controllers, detectability

Considering a linear system, we construct hybrid loops consisting in a (possibly non stabilizing) given linear dynamic continuous-time controller with a jump rule when some conditions of the state occur. This allows us to study the performance improvement problem. More precisely, we firstly show the interest of an hybrid loop to maximize the decay rate when the gain of the controller is limited. Secondly, we show how adding a hybrid loop can be useful to reduce the overshoot of an output. The proposed approaches are illustrated on two examples.

INTRODUCTION

Even for linear control systems which follow a purely continuous dynamics, it may be useful to consider dynamic controllers having a mixed discrete/continuous dynamics. This leads to the class of hybrid control laws, and the closed-loop system turns out to be a hybrid system. Such controllers are now instrumental in many feedback control designs, for their capability to provide asymptotic stability of the closed-loop system (see e.g. [START_REF] Hespanha | Stabilization of nonholonomic integrators via logic-based switching[END_REF]; [START_REF] Hespanha | Hysteresis-based switching algorithms for supervisory[END_REF]). They are also interesting for their capability to guarantee a robustness with respect to small errors in the loop, which cannot be obtained using classical (i.e. with a continuous dynamics) controllers (see e.g. [START_REF] Prieur | Asymptotic controllability and robust asymptotic stabilizability[END_REF]; Goebel and Teel (2009)).

Hybrid controllers are also instrumental to improve the performance for linear systems in presence of disturbances. See [START_REF] Chen | Analysis of reset control systems consisting of a FORE and second order loop[END_REF]; [START_REF] Beker | Fundamental properties of reset control systems[END_REF]; [START_REF] Nesić | Stability properties of reset systems[END_REF], where reset controllers are used to decrease the L 2 -gain between perturbations and the output. Consider also [START_REF] Aangenent | An LMI-based L2 gain performance analysis for reset control systems[END_REF]) (resp. Witvoet et al. (2007)) where it is shown that reset controllers may be useful to improve the L 2 (resp. H 2 )-stability of linear systems. See [START_REF] Loquen | Stability analysis for reset systems with input saturation[END_REF] for further results on reset systems with saturation in the input.

The aim of this paper is to design new hybrid strategies to improve the performance of the closed-loop system. More precisely, for linear control systems this hybrid loop may be connected to the maximization problem of the decay rate with a limited controller gain. Moreover we show that adding hybrid loops yields a solution to the reduction of the overshoot of an output, by solving a convex optimization problem. This part of our work is related to [START_REF] Haddad | Non-linear impulsive dynamical systems[END_REF] (see also Haddad et al. (2001a), and[START_REF] Haddad | Energy-based control for hybrid port-controlled Hamiltonian systems[END_REF] for Hamiltonian systems) where Lyapunov conditions are designed for the feedback interconnections of impulsive nonlinear systems. However in our paper, an a priori Lyapunov-like function is used for the designs of the hybrid controllers.

These performance results are based on the derivation of a sufficient condition for the closed-loop system with the hybrid loop to be asymptotically stable. This stability analysis relies on a LaSalle invariance principle argument and on a result of [START_REF] Prieur | Guaranteed stability for nonlinear systems by means of a hybrid loop[END_REF]. In this latter work, some a priori Lyapunov-like functions are used to design hybrid loops that stabilize the whole state or the state variable only.

The remaining part of the paper is organized as follows. The problems under consideration in this paper are introduced in Section 2. Some preliminaries are also given in this section. In Section 3, we show how we can guarantee the stability by adding a hybrid loop, this is our first main result. Section 4 contains the two other main results, namely the maximization of the decay rate with a limited jumping gain (see Section 4.1), and the reduction of the overshoot by the design of a hybrid loop (see Section 4.2). Some numerical examples illustrate the latter two results. Some concluding remarks and open questions are given in Section 5. Due to space limitation, all proofs are omitted.

Notation. The Euclidian norm is denoted by | • | and the scalar product by •, • . For positive real integers n and m, I n (resp. 0 n,m ) denotes the identity matrix (resp. the null matrix) in R n×n (resp. in R n×m ). The subscripts may be omitted when there is no ambiguity. Moreover, for a vector x, the diagonal matrix defined by the entries of x is noted diag(x), and for a matrix M , He(M ) = M + M ′ , where M ′ denotes the transpose matrix of M . For any symmetric matrix, ⋆ stands for a symmetric term.

PROBLEM STATEMENT AND PRELIMINARIES

Consider the following linear plant: ẋp = Āp x p + Bp u, y = Cp x p + Dp u, (1) with x p ∈ R np , in feedback interconnection with a (not necessarily stabilizing) linear dynamic controller: ẋc = Āc x c + Bc y, u = Cc x c + Dc y, (2) with x c ∈ R nc . Assuming that I -Dp Dc is nonsingular, the closed-loop is well posed and described by the following linear system

ẋ = Ax := A p B p B c A c x , (3)
where x = (x ′ p , x ′ c ) ′ and A p , B p , A c , and B c are matrices of appropriate dimensions uniquely defined based on the matrices in (1) and (2). In the reminder of the paper, we consider directly (3) and we do not consider anymore from which this linear system has been computed (in particular (1) and (2) will not be used anymore).

By exploiting the properties of Lyapunov functions, the aim of this paper is to construct a hybrid closed-loop system which follows the flow dynamics (3) when the state is in a set (called the flow set) and follows a suitable discrete dynamics when the state is in an other set (called the jump set). These flow and jump sets, together with the discrete dynamics, define a hybrid system and have to be designed to guarantee some performance properties of the closed-loop state x and of the plant state x p .

More precisely, we will address the following two problems. The first problem is an optimization of the decay rate of the variable x p by adding a hybrid loop to the system (3). Problem 1. Consider the closed-loop system (3). Given a limitation on the amplitude of the gain K, design a hybrid system which follows the dynamics (3) when flowing, follows the dynamics

x + = Kx when jumping, and is such that the x p -variable have the best decay rate of the x p -variable as possible.

The second problem is related to the reduction of the overshoot of the output y of (1) by means of a hybrid loop. Problem 2. Consider the closed-loop system (3). Design a hybrid system which follows the dynamics (3) when flowing and the output y of (1) converges to zero and is non-increasing along solutions.

Both problems are solved in Theorems 2 and 3 respectively. Note that Problem 2 is solved indirectly. More precisely, to solve it, we introduce a function V p : R np → R ≥0 , and we design a hybrid loop such that V p is non-increasing along the solutions of the hybrid system, and V p (x p ) is close (in a certain manner) to |y| 2 . See Theorem 3 for more details.

Let us shortly describe the framework of the hybrid systems that is considered in this paper. For an introduction, see e.g. the recent survey Goebel et al. (2009). Such a system combines a continuous dynamics in a set F (called flow set) and discrete dynamics in a set J (called jump set), and it is formally written as

ẋ = Ax if x ∈ F , x + = Kx if x ∈ J , (4)
where

x = (x ′ p , x ′ c ) ′ ∈ R n , n = n p + n c
, A and K are two matrices of appropriate dimensions. We recall some basic ingredients on hybrid system theory, and solutions of (4). Due to mixed discrete/continuous dynamics, a solution of (4) will be defined on a mixed discrete/continuous time domain. More precisely, a set E is a hybrid time domain if for all (T 0 , J

0 ) ∈ E, E ∩ ([0, T 0 ] × {0, 1, . . . J 0 }) is a compact hybrid time domain, i.e.

it can be written as

J-1 j=0 ([t j , t j+1 ], j), for some finite sequence of times 0 = t 0 ≤ t 1 . . . ≤ t J . A solution x to (4) consists of a hybrid time domain dom x and a function x : dom x → R n such that x(t, j) is absolutely continuous in t for a fixed j and (t, j) ∈ dom x satisfying (S1) for all j ∈ N and almost all t such that (t, j) ∈

dom x, x(t, j) ∈ F, ẋ(t, j) = Ax(t, j) ; (S2) For all (t, j) ∈ dom x such that (t, j +1) ∈ dom x,
x(t, j) ∈ J, x(t, j + 1) = Kx(t, j) .

Then, the state solution x is parameterized by (t, j) where t is the ordinary time and j is an independent variable that corresponds to the number of jumps of the solution.

When the state x(t, j) belongs to the intersection of the flow set and of the jump set, then the solution can either flow or jump. This parameterization may be omitted when there is no ambiguity.

A solution x to (4) is said to be complete if dom x is unbounded, Zeno if it is complete but the projection of dom x onto R ≥0 is bounded, and maximal if there does not exist an other solution x of (4) such that x is a truncation of x to some proper subset of dom x. Hereafter, only maximal solutions will be considered. For more details about this hybrid systems framework, we refer the reader to Goebel et al. (2009) or [START_REF] Prieur | Hybrid feedback control and robust stabilization of nonlinear systems[END_REF]. Definition 1. The hybrid system ( 4) is said to be

• stable: if for each ǫ > 0 there exists δ > 0 such that each solution x to (4) with |x(0, 0)| ≤ δ satisfies |x(t, j)| ≤ ǫ for all (t, j) ∈ dom x; • attractive: if every solution x to (4) is complete and satisfies lim t+j→∞ |x(t, j)| = 0; • globally asymptotically stable: if it is both stable and attractive.

GUARANTEEING ASYMPTOTIC STABILITY

Before solving Problems 1 and 2, we first state a result suggesting a sufficient condition for the hybrid system (4) to be globally asymptotically stable. To do that, some Lyapunov functions are used and since the closed-loop system (3) is linear, we restrict to the class of quadratic functions, and to sets F and J depending on the sign of a quadratic function of the state.

Moreover since the x p -variable is the state of the plant, it is natural to impose the following structure on the gain K defining the discrete dynamics in (4):

K = I p 0 K p 0 ,
where K p is a given matrix in R nc×np .

More precisely given a symmetric positive definite matrix Pp in R n×n , a matrix N in R n×n , a matrix K p in R nc×np , and a positive value ᾱ, we will consider the following structure for the hybrid system (4):

ẋ = Ax if x ∈ F , (x + p , x + c ) = (x p , K p x p ) if x ∈ J , (5)
where F ⊂ R n and J ⊂ R n are the closed subsets of R n defined by

F = {x ∈ R n , x ′ N x ≤ -ᾱx ′ p Pp x p } , J = {x ∈ R n , x ′ N x ≥ -ᾱx ′ p Pp x p } , (6)
where N , ᾱ, Pp will be specified in the sequel.

Our first main result is based on the following proposition which gives a convergence property of the x p -variable only: Proposition 1. [START_REF] Prieur | Guaranteed stability for nonlinear systems by means of a hybrid loop[END_REF]] Consider the closedloop system (3) and a symmetric positive definite matrix np+nc) such that the matrix

P = P p P pc ⋆ P c in R (np+nc)×(
Pp := P p -P pc P -1 c P ′ pc satisfies He( Pp (A p + B p K p )) < -α Pp , (7)
for some α > 0 and for K p := -P -1 c P ′ pc . Then, letting

N := He Pp A p Pp B p 0 0 . ( 8
)
the hybrid system ( 5), ( 6) is such that the plant state x p converges to zero, and x p → V p (x p ) := x ′ p Pp x p is nonincreasing along solutions.

With Proposition 1, there is no guarantee that the state of the controller will converge to zero. It turns out to be sufficient that the controller dynamics is detectable from the output matrix B p . Intuitively, this requirement corresponds to asking that any nonzero controller evolution will be detected by the plant states so that a LaSalle result can be applied to show convergence. This is the basic tool to prove the following theorem. Theorem 1. Consider any state stabilizing gain K p such that A p + B p K p is Hurwitz and a pair Pp , α satisfying (7). Consider now any ᾱ ≤ α and the flow and jump sets in (6). If the pair (B p , A c ) is detectable, then the hybrid system ( 5) is globally exponentially stable.

IMPROVING THE PERFORMANCE

In this section, we rely on the results of Theorem 1 to propose a suitable jump rule to be incorporated in a linear continuous-time control system to maximize the decay rate or to reduce the plant output overshoot (see respectively Sections 4.1 and 4.2 below). This will solve Problems 1 and 2.

Maximizing the decay rate

Consider equation ( 7) and note that it resembles the classical state-feedback stabilization problem with guaranteed convergence rate, which is well known to be solvable via a generalized eigenvalue problem. Given a plant, a possible way to address the search for Pp , K p and α satisfying ( 7) is to impose a bound κ M on the size of K p and compute the maximum α satisfying (7) (or a conservative estimate of it) while guaranteeing |K p | ≤ κ M . This type of goal is achieved by solving the following generalized eigenvalue problem:

Find a symmetric positive matrix Qp in R np×np , a matrix X in R nc×np and a positive value α solution of

max Qp= Q′ p ,α,X α, s.t. Qp ≥ I 0 ≤ κ M I X X ′ κ M I -α Qp ≥ He(A p Qp + B p X), (9)
where κ M > 0 is given. The optimal solution to (9) will lead to the gain K p = X Q-1 p and to Pp = Q-1 p satisfying (7) and such that

|K p | ≤ |X||Q -1 p | ≤ κ M because the first constraint imposes |Q -1
p | ≤ 1 and the second constraint imposes |X| ≤ κ M . Using this gain, we may solve Problem 1 and the following theorem holds Theorem 2. Consider a symmetric positive matrix Qp in R np×np , a matrix X in R nc×np and a positive value α satisfying (9). Let K p = X Q-1 p and Pp = Q-1 p . For any selection of 0 < ᾱ ≤ α and the flow and jump sets in ( 6), then 1. for a suitable k > 0, the plant state response satisfies the following exponential bound for all (t, j) ∈ dom x:

|x p (t, j)| ≤ k exp - ᾱ 2 t |x p (0, 0)| 2. if the pair (B p , A c
) is detectable, then the hybrid closed-loop is globally exponentially stable.

The optimization problem in (9) can be used to build a curve providing the maximum decay α achievable with a certain bound κ M on the gain K p . Then in many practical cases, it might well be that the optimal α grows unbounded as the bound κ M becomes arbitrarily large. This fact is illustrated in the next example study.

Example 1. Consider a one-dimensional linear plant connected in negative feedback with a one-dimensional linear controller: ẋp = a p x p + b p x c ẋc = a c x cx p , where we assume b p > 0.

For this example, the solution to (9) can be computed explicitly as a function of κ M and corresponds to K p = -κ M , while the optimal performance is α = -2a p +2b p κ M , achieved with Qp = 1 and X = -κ M . It is instructive to study the shape of the jump and flow sets F and J defined in (6), which are symmetric cones in the plane (x p , x c ). Using, for example, ᾱ = α 2 = -a p + b p κ M , the sets in (6) become

F = (x p , x c ) : [ xp xc ] ′ a p + b p κ M b p b p 0 [ xp xc ] ≤ 0 , ( 10
)
while J is the closure of the complement of F.

According to the results in Theorem 2 the hybrid closedloop system given by ẋp = a p x p + b p x c ẋc = a c x cx p , if x ∈ F,

x + p = x p x + c = -κ M x p , if x ∈ J
is globally exponentially stable.

The flow set in (10) defines a symmetric cone in the (x p , x c ) plane whose boundaries correspond to the two subspaces {(x p , x c ) : Some trajectories are also reported in Figure 1, corresponding to the case a c = 0 which, in the case κ M = 1 corresponds to a generalized version of the behavior of the so-called Clegg integrator (see [START_REF] Clegg | A nonlinear integrator for servomechanisms[END_REF]; [START_REF] Zaccarian | First order reset elements and the Clegg integrator revisited[END_REF]) in negative feedback interconnection with the plant P(s) = 1 s+1 . The generalization consists in resetting the Clegg integrator state to the value of the input rather than to zero but essentially coincides with the original resetting rule where the integrator was supposed to be reset only at zero crossings of the integrator input (see [START_REF] Clegg | A nonlinear integrator for servomechanisms[END_REF]; [START_REF] Zaccarian | First order reset elements and the Clegg integrator revisited[END_REF] for details). Note that in the case κ M = 2 reported to the rigth of Figure 1, the horizontal boundary of the flow set is tilted, reducing the size of the flow set, and the reset manifold x c = K p x p is tilted as well, to speed up the state decay rate. Nevertheless, the vertical boundary corresponding to {(x p , x c ) : x p = 0}, remains unchanged thus preserving the peculiarity that the response exhibits no overshoot. This property arises from the fact that the function x p → x 2 p is nonincreasing along the solutions. This fact is further illustrated by a few simulations starting from the initial conditions (x p (0, 0), x c (0, 0)) = (-1, 0), reported at the bottom plot of Figure 1. In both simulations, the controller state is first reset to the reset manifold and then exhibits a flow interval until the state reaches the vertical axis, where the solution is finally reset to zero. ⋆

x p = 0},

Overshoot reduction

Perhaps the first example study where hybrid control systems were shown to overcome intrinsic limitations of classical linear control was the one in [START_REF] Beker | Plant with integrator: an example of reset control overcoming limitations of linear feedback[END_REF], were a reset controller was shown to improve upon linear control in terms of overshoot reduction. If the plant ( 1) is strictly proper (namely, Dp = 0), a possible way to mathematically formalize the requirement of overshoot reduction in terms of the Lyapunov functions introduced here is to construct a quadratic function x p → V p (x p ) := x ′ p Pp x p and the stabilizing gain K p in such a way that, to a certain extent, V p (x p ) ≈ |y| 2 . Then achieving non-increase of V p via the hybrid loops of this paper will induce (almost) no overshoot, namely |y(t)| is (almost) non-increasing along trajectories. Let us make more explicit this informal discussion.

Finding the above discussed pair (V p , K p ) can be done, for example, by writing the dynamics (1) in observability canonical form, so that x p = (x ′ 1 , y) ′ and solving the following LMI eigenvalue problem: Find a symmetric positive matrix Qp in R np×np , a matrix X in R nc×np and positive value α, ρ x , and ρ y solution of

min Qp= Q′ p ,ρx,ρy,X ρ y , s.t. Qp = Q1 q1y q ′ 1y qy > I 0 < κ M I X X ′ κ M I 0 > He(A p Qp + B p X) 0 ≤ ρ x 1 1 ρ y ρ x < Q 1 q y < 1 + ρ y , (11)
where κ M > 0 is given. Similar to the case addressed in Section 4.1, the optimal solution to (11) will lead to the gain K p = X Q-1 p and to Pp = Q-1 p satisfying (7) with a small enough α and such that |K p | ≤ |X|| Q-1 p | ≤ κ M . Moreover, the bounds given by the last three constraints can be shown to imply that smaller values of ρ y will lead to a function

V p (x p ) = x ′ p Q-1 p x p closer
→ 0 as k → ∞. Then, defining V k p (x p ) = x ′ p Qk p -1 x p , we have, for each x p in R np , lim k→∞ V k p (x p ) = |y| 2 , ( 12
)
namely as ρ k y approaches zero, V k p (x p ) approaches |y| 2 . Moreover, given any pair ( Qp , X) satisfying ( 11), let K p = X Q-1

p and Pp = Q-1 p . Then, for a small enough selection of α, equation ( 7) is satisfied. Moreover, given any selection of 0 < ᾱ ≤ α, the reset system ( 5) with the flow and jump sets in ( 6), is such that:

1. the plant state x p converges to zero, and the function x p → V p (x p ) = x ′ p Pp x p is non-increasing along solutions; 2. if the pair (B p , A c ) is detectable, then the hybrid closed-loop is globally exponentially stable.

Just as in the case discussed in Section 4.1, the optimization problem in ( 9) can be used to build a curve providing the overshoot reduction level ρ -1 y achievable with a certain bound κ M on the gain K p . Similar to before, it might well be that the optimal ρ -1 y grows unbounded as the bound κ M becomes arbitrarily large. Example 2. We consider an example originally discussed in [START_REF] Beker | Plant with integrator: an example of reset control overcoming limitations of linear feedback[END_REF], where a First Order Reset Element whose linear part is characterized by the transfer function 1 s+1 controls via a negative unitary feedback a SISO plant whose transfer function is P = s+1 s(s+0.2) . For this example, the control system involving the FORE is shown in [START_REF] Beker | Plant with integrator: an example of reset control overcoming limitations of linear feedback[END_REF] to behave more desirably than the linear control system because it has only about 40% overshoot as compared to the linear closed-loop system, while retaining the rise time of the linear design (this example was also later discussed later in [START_REF] Nesić | Stability properties of reset systems[END_REF]; Zaccarian et al. (to appear) where the L 2 gain properties of the reset closed-loop were characterized). Here we show that when allowing more general resets than just the ones induced by the FORE resetting law (which essentially imposes a reset of the controller state to zero when the controller input and output have opposite signs), arbitrary small overshoot can be achieved, while retaining the same rise time. To this aim, we use the following state-space representation of the linear closed-loop:

A p B p B c A c =   -0.6 0.6 -1 -0.4 0.4 1 0 1 -1   . ( 13
)
Figure 2 shows the overshoot reduction parameter ρ -1 y as a function of the bound κ M imposed on the gain K p . Note that the curve has a peculiar shape for small values of κ M and then essentially grows linearly or large values of κ M . Table 2 reports the values of Pp , K p and ρ -1 y for some selections of κ M . Using the values in Table 2, we run several simulations of the closed-loop starting from the initial conditions x p (0, 0) = [1, 1] ′ , resembling the In particular, we implement the hybrid loops arising when using the values in the table within the scheme proposed in Theorem 3. Note that since in this case the pair (B p , A c ) is observable, then according to the results of Theorem 1, the trajectories are all convergent to zero. The resulting plant input and output responses are shown in Figure 3, where we also show the linear response (thin solid) and the response obtained with the FORE used in [START_REF] Beker | Plant with integrator: an example of reset control overcoming limitations of linear feedback[END_REF] (thin dashed). The results in Figure 3 are indicative of the potential of the proposed approach with the goal of reducing overshoots in linear control systems and correspond to trajectories of the reset system (5) with the flow and jump sets in (6), with the parameters in (13) and with the gains K p in Table 2. It should be pointed out that even though we manage to improve the overshoot reduction as compared to the FORE resetting strategy proposed in [START_REF] Beker | Plant with integrator: an example of reset control overcoming limitations of linear feedback[END_REF] (see the thin dashed curve in Figure 3), this last strategy is more appealing from an implementation viewpoint because it only requires a measurement of the plant output. Conversely, the resetting strategy of the improved bold curves of the figure are obtained using a full measurement of the plant state. ⋆

CONCLUSIONS

The design problem of a stabilizing hybrid loop is considered in this paper. This class of system mixes discrete and continuous dynamics depending on the value of a quadratic function of the state. Adding a hybrid loop to a closed-loop linear system can been instrumental when maximizing the decay rate with an amplitude-limited controller. Finally it has been proven that adding a hybrid loop may be useful to reduce an overshoot of a given output. Some simulations highlighted the interest of the results.

Fig. 1 .

 1 Fig. 1. Shapes of the flow and jump sets of Example 1 using κ M = 1 (upper left) and κ M = 2 (upper right). The lower plots show the time histories of the two trajectories shown on the state space planes.

  namely the vertical axis, and {(x p , x c ) : 2b p x c = -(a p + b p κ M )x p }, namely (by the fact that b p > 0 and κ M > 0), a line which rotates clockwise as κ M (and, consequently, the guaranteed convergence rate ᾱ = -a p + b p κ M ) increases. An example corresponding to a p = -1, b p = 1 and κ M = 1 is shown in the top left plane of Figure 1. In this case, the flow set (striped area) corresponds to the second and fourth quadrants and the set x c = K p x p = -κ M x p = -x p , where the controller state is reset, (depicted as a bold dashed line) corresponds to the bisector of the second and fourth quadrant. The top right plane in the same figure shows how these sets change when increasing the gain to κ M = 2.

Fig. 2 .

 2 Fig. 2. Curve of the overshoot reduction parameter ρ -1 y as a function of the bound κ M imposed on the gain K p . κ M Pp K ′ p

Fig. 3 .

 3 Fig. 3. Simulation results when using the linear hybrid loops discussed in Example 2. Linear response (thin solid) and response of the FORE closed-loop of Beker et al. (2001) (thin dashed) compared to the reset responses obtained with the parameters inTable 2 (bold).

  to |y| 2 . This fact, together with desirable stability properties of the hybrid closed-loop, helps us to solve Problem 2 and to establish in the following theorem. Theorem 3. Consider a sequence of solutions ( Qk p , ρ k y , ρ k x , X k ) k∈N to the constraints (11) such that ρ k y

Table 1 .

 1 Values of the matrices Pp , K p and ρ -1 y for some selections of κ M . step responses reported in Beker et al. (2001); Nesić et al. (2005); Zaccarian et al. (to appear).

		0.31315 -0.38064 -0.38064 0.78906	0.046750 -0.087302	0.48451
	0.5	0.10170 -0.088509 -0.088509 0.99128	0.055896 -0.49023	10.662
	1	0.049336 -0.045986 -0.045986 0.99778	0.057423 -0.97424	21.179
	5	0.0097785 -0.0096376 -0.0096376 0.99991	0.059499 -4.8307	103.25

1 Work supported in part by the ANR project ArHyCo, ARPEGE, contract number ANR-2008 SEGI 004 01-30011459. part by ENEA-Euratom and MIUR.


Claudia Cagnoli

Giovanni Stevanin

Alessandro Brussino

Cecilia Mancini

Russell L Margolis

Susan E Holmes

Marcello Nobili

Sylvie Forlani

Sergio Padovan

Patrizia Pappi

Marco Barberis

Cécile Zaros

Isabelle Leber

Pascale Ribai

Luisa Pugliese

Corrado Assalto

Alexis Brice

Nicola Migone

Alexandra Dürr

Alfredo Brusco
email: alfredo.brusco@unito.it

MISSENSE MUTATIONS IN THE AFG3L2 PROTEOLYTIC DOMAIN ACCOUNT FOR 1.5% OF EUROPEAN AUTOSOMAL DOMINANT CEREBELLAR ATAXIAS

Keywords: Autosomal Dominant Cerebellar Ataxia, Spinocerebellar ataxia, SCA28, AFG3L2, mutation screening Autosomal Dominant Cerebellar Ataxia, Spinocerebellar ataxia, SCA28, AFG3L2, mutation screening

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INTRODUCTION

Autosomal dominant cerebellar ataxias (ADCAs, or SCAs) are a group of clinically heterogeneous neurodegenerative disorders primarily characterized by imbalance, progressive gait and limb ataxia, and dysarthria [START_REF] Harding | Clinical features and classification of inherited ataxias[END_REF]. The clinical phenotype is often complicated by the presence of additional neurological signs, which are highly variable among and within families [START_REF] Finsterer | Ataxias with autosomal, X-chromosomal or maternal inheritance[END_REF][START_REF] Schols | Autosomal dominant cerebellar ataxias: clinical features, genetics, and pathogenesis[END_REF]. SCAs are among the most genetically heterogeneous neurodegenerative diseases, at present nearly thirty genetically distinct subtypes have been defined (http://www.neuro.wustl.edu/neuromuscular/ataxia/domatax.html). In most countries, an expansion of a coding CAG-triplet repeat, resulting in production of a protein with an abnormal polyglutamine (polyQ) stretch, accounts for 40-60% of ADCA. In some populations, as in Southern Brazil, founder effects have raised this ratio up to ~100% [START_REF] Jardim | A survey of spinocerebellar ataxia in South Brazil -66 new cases with Machado-Joseph disease, SCA7, SCA8, or unidentified disease-causing mutations[END_REF][START_REF] Storey | Frequency of spinocerebellar ataxia types 1, 2, 3, 6, and 7 in Australian patients with spinocerebellar ataxia[END_REF]. Less common SCAs are caused by other types of mutations: the expansion of a tri-or penta-nucleotide in untranslated regions (CAG in SCA12,ATTCT in SCA10,and TGGAA in SCA31), by point mutations (SCA5, SCA11, SCA13, SCA14, and SCA27), or by gene dosage anomalies (SCA15 and SCA20) [START_REF] Fahey | Spinocerebellar ataxia type 14: study of a family with an exon 5 mutation in the PRKCG gene[END_REF][START_REF] Holmes | Expansion of a novel CAG trinucleotide repeat in the 5' region of PPP2R2B is associated with SCA12[END_REF]Houlden, et al., 2007;[START_REF] Ikeda | Spectrin mutations cause spinocerebellar ataxia type 5[END_REF][START_REF] Knight | A duplication at chromosome 11q12.2-11q12.3 is associated with spinocerebellar ataxia type 20[END_REF][START_REF] Matsuura | Spinocerebellar ataxia type 10: a disease caused by a large ATTCT repeat expansion[END_REF][START_REF] Sato | Spinocerebellar ataxia type 31 is associated with "inserted" penta-nucleotide repeats containing (TGGAA)n[END_REF][START_REF] Van De Leemput | Deletion at ITPR1 underlies ataxia in mice and spinocerebellar ataxia 15 in humans[END_REF][START_REF] Waters | Mutations in voltage-gated potassium channel KCNC3 cause degenerative and developmental central nervous system phenotypes[END_REF]. Most recently, point mutations of AFG3L2 (OMIM #610246) have been shown to cause SCA28 [START_REF] Bella | Mutations in the mitochondrial protease gene AFG3L2 cause dominant hereditary ataxia SCA28[END_REF][START_REF] Edener | Early onset and slow progression of SCA28, a rare dominant ataxia in a large four-generation family with a novel AFG3L2 mutation[END_REF]. AFG3L2 was originally cloned as a paralogue of the SPG7 gene (encoding for paraplegin, OMIM *602783) [START_REF] Banfi | Identification and characterization of AFG3L2, a novel paraplegin-related gene[END_REF], whose loss-of-function causes an autosomal recessive form of hereditary spastic paraplegia (HSP) [START_REF] Casari | Spastic paraplegia and OXPHOS impairment caused by mutations in paraplegin, a nuclear-encoded mitochondrial metalloprotease[END_REF]. AFG3L2 encodes for a subunit of the hetero-oligomeric m-AAA protease (ATPases associated with various cellular activities), a component of the mitochondrial ATP-dependent metalloprotease located on the inner mitochondrial membrane. The AAA metalloproteases take part in proteolytic quality control and chaperon-like activities in mitochondria by degrading misfolded proteins and promoting the assembly of respiratory chain complexes [START_REF] Leonhard | Chaperone-like activity of the AAA domain of the yeast Yme1 AAA protease[END_REF].

Here we examine the prevalence of SCA28 among Caucasian ADCA families, further defining the

PATIENTS AND METHODS

Patients

We recruited 366 index cases with progressive cerebellar ataxia and a family history of a similar disorder (defined as the presence of at least two affected individuals in at least two consecutive generations). The mean age at onset was 41.1 ± 18.3 years (range: 1-79 years; onset defined as the year of the first symptoms, as reported by the patient or the family). When possible, additional family members were recruited if a mutation in AFG3L2 was found in a proband. Most patients originated from France (n= 240), but the collection also included patients from other countries in Europe (n= 50), the United States (n =66, predominately of European origin), North-Africa/Middle-East (n= 6), French West-Indies (n= 3) and Madagascar (n= 1). We excluded individuals with pathogenic expansions in the SCA1-3, 6, 7, 17 and DRPLA loci, or recurrent mutations in the SCA5, SCA13, SCA14, or FGF14/SCA27 loci (Klebe, et al., 2005) (Stevanin et al., unpublished data). Clinical and genetic studies were performed after obtaining informed consent from all participants or the parents of participating minors and with the approval of the local ethics committees. Ninety-five French and 95 Italian healthy controls were examined to establish the frequency of variants in the normal population.

Mutation screening

The 17 coding exons of the AFG3L2 gene (RefSeq NM_006796.1) were PCR amplified using primer and conditions reported in Supp. materials and methods and Supp. Table S1. Mutation analysis was performed on the amplicons using a DHPLC WAVE System (Transgenomics) with melting temperatures (Tm and Tm+2°C) determined by DHPLC Melt software [START_REF] Jones | Optimal temperature selection for mutation detection by denaturing HPLC and comparison to single-stranded conformation polymorphism and heteroduplex analysis[END_REF] (Supp. Table S1). A normal control profile was always compared with that from a patient. PCR products showing a DHPLC peak shift were purified using the ExoSAP method (MBI-Fermentas, 1.1 and an ABI Prism 3100 Avant automatic sequencer (Applied Biosystems, Foster City, CA). The 17 coding exons of the SPG7 gene (RefSeq NM_003119.2) were also amplified and directly sequenced following primers and conditions reported in Supp. materials and methods and Supp. Table S2.

AFG3L2 gene copy number was evaluated by quantitative duplex PCR (qPCR) using a Roche-UPL assay centered on exon 14 (Roche-Diagnostics, Mannheim, Germany). The gene dosage strategy was based on the relative amplification of the target sequence (AFG3L2) and the co-amplified internal standard RNaseP using the comparative delta Ct method described elsewhere [START_REF] Livak | Analysis of relative gene expression data using real-time quantitative PCR and the 2(-Delta Delta C(T)) Method[END_REF]) (see Supp. materials and methods).

Nucleotide numbering throughout the paper follows cDNA numbering: +1 refers to the first nucleotide of the ATG translation initiation codon of the corresponding RefSeq, according to journal guidelines (www.hgvs.org/mutnomen). The initiation codon is codon 1.

Haplotype reconstruction

Five 6-FAM fluorescently-labelled microsatellites spanning ~10 Mb on chromosome 18, including two markers telomeric (D18S1150 and D18S53) and three centromeric (D18453, D18S1104, and D18S1107) to AFG3L2 were genotyped in all available subjects with a specific mutation whenever the mutation occurred in at least two unrelated families. We used standard PCR conditions and primers described at www.genome.ucsc.edu. Haplotypes were manually reconstructed.

Homology modeling of AFG3L2, in-silico

The three-dimensional model of human AFG3L2 protein was generated using "homology modelling", a bioinformatics algorithm that builds a model of the "target" protein based on the homology of its amino acid sequence with that of proteins of known structure [START_REF] Marti-Renom | Comparative protein structure modeling of genes and genomes[END_REF]. The AFG3L2 protein model was constructed with the NEST application (Petrey, et al 6 2003), using the T. maritima FtsH 2CE7 crystal structure as a template (see Supp. materials and methods). Mutations p.Thr654Ile, p.Met666Val, p.Met666Thr, p.Met666Arg, p.Gly671Glu and p.Gly671Arg were introduced into the model with the Yasara software (http://www.yasara.org) and their consequences were evaluated using a series of bioinformatics tools (see also Supp. materials and methods).

Multiple species alignment of AFG3L2 protein was made with the ClustalW software (http://www.ebi.ac.uk/Tools/clustalw2/index.html) using orthologue sequences obtained through the Ensembl genome browser (http://www.ensembl.org). In silico analysis of the possible pathogenicity of each amino acid substitution was performed with two different applications: (i) PolyPhen ("Polymorphism Phenotyping", http://genetics.bwh.harvard.edu/pph/index.html), selecting the Protein Data-Base (PDB) as a source for sequence alignment, and (ii) SIFT ("Sorting Intolerant From Tolerant", http://blocks.fhcrc.org/sift/). Possible effects on splicing were checked using the applications Splice Site Prediction (http://www.fruitfly.org/seq_tools/splice.html) [START_REF] Reese | Improved splice site detection in Genie[END_REF], ESEfinder 2.0 (http://rulai.cshl.edu/tools/ESE/) [START_REF] Cartegni | ESEfinder: A web resource to identify exonic splicing enhancers[END_REF][START_REF] Smith | An increased specificity score matrix for the prediction of SF2/ASF-specific exonic splicing enhancers[END_REF], and PESXs (http://cubweb.biology.columbia.edu/pesx/) [START_REF] Zhang | Exon inclusion is dependent on predictable exonic splicing enhancers[END_REF]. One variant, predicted to alter splicing, was further investigated using a minigene assay (see Supp. materials and methods).

RESULTS

SCA28 mutations in ADCA families.

Three-hundred sixty six unrelated patients with ADCA were screened for point mutations in the AFG3L2 gene using DHPLC, followed by direct sequencing of the amplimers with shifted peaks.

We found six missense changes in nine unrelated index cases: eight were French and one was of Italian origin (9/366, 2.6%): c.1961C>T (p.Thr654Ile) in exon 15, c.1996A>G (p.Met666Val), c.1997T>G (p.Met666Arg), c.1997T>C (p.Met666Thr), c.2011G>A (p.Gly671Arg) and c.2012G>A (p.Gly671Glu) in exon 16 (Figure 1). These mutations were not reported as 7 polymorphisms in the dbSNP database build 130 (http://www.ncbi.nlm.nih.gov/projects/SNP/), and were not found among 380 French or Italian healthy control chromosomes. DNA from 27 additional family members in six families was available, which allowed us to verify that the mutation segregated with the disease and was absent in healthy relatives (Figure 2). The mutations were located in the peptidase-M41 domain of the AFG3L2 protein. Multiple species alignment showed that the three amino acids altered by mutations (Thr654, Met666 and Gly671) are conserved through Saccharomyces cerevisiae (Figure 1). The applications PolyPhen and SIFT predicted that all the amino acid mutations are deleterious. Three mutations (p.Thr654Ile, p.Met666Val and p.Gly671Arg) were found in two families each. Haplotype reconstruction showed that identical mutations shared a common haplotype, suggesting that the families have a common ancestor (Figure 2).

In addition to the six missense mutations, we found 25 variants, 14 of which were neither reported as polymorphisms in the dbSNP database nor found in healthy controls (Supp. Table S3). Twelve were intronic nucleotide changes and two were synonymous substitutions. In silico analysis of the effect of each variant on splicing showed that c.293-13_293-14delTT changes the score of the exon 4 acceptor splice site from 0.22 to 0.03. Neither relatives nor cDNA of this patient were available for further study. We tested the effect of this mutation using a minigene assay, which showed it has no effect on splicing.

We also excluded whole-gene deletion/duplication of AFG3L2 in 129/366 subjects (35%) by qPCR of exon 14, and deletions in 171 patients, who carried one or more heterozygous SNP at DHPLC and / or direct sequencing.

3D reconstruction of AFG3L2 and predicted effect of the mutations on intermolecular electrostatic parameters.

The three-dimensional reconstruction of the C-terminal region of AFG3L2, the homohexameric complex, and the relative positions of mutated amino acids are depicted in Figure 3A-B this model, the amino acids Met666 and Gly671 are always on the surface of the complex, regardless of the conformation analyzed, suggesting that these two residues may be involved in interactions with other molecules. p.Gly671Glu significantly increases the electrostatic potential difference between the inner-mitochondrial-membrane side and the matrix side of the hexamer, whereas p.Met666Arg and p.Gly671Arg decreased it (Figure 3C and Supp. Table S4). The other three mutations do not significantly affect the potential difference (Supp. Table S4). Given that the central pore of the m-AAA complex (Figure 3D) is used to convey substrates to the proteolytic chamber [START_REF] Kress | The alternating power stroke of a 6-cylinder AAA protease chaperone engine[END_REF], we investigated the effect of the mutations of the amino acids close to the central pore. As shown in Figure 3E, 3F and Supp. Table S4, p.Gly671Glu increases the central pore dipole energy, and p.Met666Arg and p.Gly671Arg decreased it. In addition, we found that all the mutations analyzed decreases the mean interaction energy of the hexamer, destabilizing the m-AAA complex (Supp. Table S5). Taken together, the bioinformatics data point to a generalized destabilizing effect of the six mutations detected in our patient population, with p.Met666Arg, p.Gly671Glu, and p.Gly671Arg also affecting the charge of the hexamer and, in particular, the charge of the translocation channel.

SPG7 mutation analysis

Since AFG3L2 and paraplegin can interact to form the multimeric m-AAA protease complex, we speculated that SPG7 variants could modify the SCA28 phenotype. We screened for SPG7 variants by direct sequencing the coding exons of nine SCA28 patients with the earliest onset (AAD-080_9, AAR-197_1, SAL-872_6, SAL-872_9, SAL-331_11, AAD-444_3, AAD-444_9, AAD-455_19, AAD-701_21). Two unreported intronic variants were detected: c.1678+13 C>T in subject 701_21; c.2196+18 C>G in subject 331_11. Neither variant is predicted to alter splicing. 9 families, 14 males and 11 females, clinically examined at a mean age of 45.4 ± 17.7 years (range 10-77 yrs). The mean age at onset was 30.7 ± 16.2 yrs (range: 6-60 yrs). Cerebellar gait ataxia was the initial clinical abnormality in all but one patient (AAR-197_8). At examination, cerebellar ataxia was associated with dysarthria (17/25, 68%), ophthalmoplegia (12/25, 48%) and/or gazedevoked nystagmus (13/24, 54%). In addition, saccadic pursuit was noted in 6/16 (37%) and slow saccades were present in 3/12 (25%). Ptosis was often present (8/19, 42%) and it was not correlated with disease duration (p= 0.937). Summing up gazed-evoked nystagmus, ophthalmoplegia and ptosis, 19 out of 25 patients had one or more oculomotor anomalies (76%). Six patients had a full pyramidal syndrome (i.e., increased reflexes and Babinski sign) and gait spasticity was evident in one patient (AAR-197_10), two features already described in SCA28 [START_REF] Mariotti | Spinocerebellar ataxia type 28: a novel autosomal dominant cerebellar ataxia characterized by slow progression and ophthalmoparesis[END_REF].

Clinical features

Interestingly, extrapyramidal signs, either dystonia (n=3) or parkinsonism (n=3), were also present although relatively infrequent (24%). In patient SAL-872_6, the etiology of parkinsonism may have been related to concomitant treatment with a neuroleptic. Vibration sense at the ankles was decreased in 9/20 patients (45%), but nerve conduction studies, available for seven patients (five of whom had vibration sense alterations) were normal, with neurogenic changes on electromyography observed only in SAL-331_4 and SAL-331_11. Two patients from the same family (SAL-331) had low verbal IQ (78 and 89, respectively). Cerebral MRI (n=10) or CT scan (n=2) revealed cerebellar atrophy affecting predominantly the vermis. The brainstem was normal. A muscle biopsy performed in one patient was unremarkable.

Overall, the disease was slowly progressive and rarely severe. Only patient SAL-444_3 needed help for walking, 29 years after the onset of symptoms, and five clinically affected patients of two families were not functionally incapacitated (SAL-701_29, SAL-701_021, AAD-080_56, AAD-080_57 and AAD-080_58).

DISCUSSION

The m-AAA protease is an ATP-dependent proteolytic complex located in the matrix side of the 10 inner mitochondrial membrane and involved, in yeast, in protein quality control and protein processing [START_REF] Arlt | The YTA10-12 complex, an AAA protease with chaperone-like activity in the inner membrane of mitochondria[END_REF][START_REF] Esser | A novel two-step mechanism for removal of a mitochondrial signal sequence involves the mAAA complex and the putative rhomboid protease Pcp1[END_REF][START_REF] Leonhard | Membrane protein degradation by AAA proteases in mitochondria: extraction of substrates from either membrane surface[END_REF]. Human m-AAA protease consists of two isoenzymes: (i) a hetero-oligomeric complex formed by paraplegin (SPG7) and AFG3L2, and (ii) a homo-oligomeric AFG3L2 complex. Loss-of-function mutations in the SPG7 gene cause an autosomal recessive hereditary spastic paraparesis (HSP), whereas AFG3L2 mutations cause SCA28 [START_REF] Bella | Mutations in the mitochondrial protease gene AFG3L2 cause dominant hereditary ataxia SCA28[END_REF][START_REF] Edener | Early onset and slow progression of SCA28, a rare dominant ataxia in a large four-generation family with a novel AFG3L2 mutation[END_REF]. In 366 unrelated ADCA patients of predominately European ancestry, we found nine affected subjects harboring six different AFG3L2 mutations, clustered in exons 15 (p.Thr654Ile) and 16 (p.Met666Arg, p.Met666Thr, p.Met666Val, p.Gly671Arg and p.Gly671Gln). The pathogenicity of these missense substitutions is supported by multiple lines of evidence: (i) in families where more than one affected subject was available, we demonstrated that the mutation segregated with the disease and was not present in healthy subjects of the family. In one of these families (AAD-080), a positive multipoint LOD score of 2.54 at D18S853 highly suggested its linkage to the SCA28 locus (data not shown).

Moreover, STR-analysis revealed that patients from different families harboring the same mutation shared a common haplotype, suggesting the presence of common ancestors. (ii) Mutations were not found in 380 control chromosomes. (iii) The mutated amino acids are evolutionary conserved up to the S. cerevisiae orthologous protein Yta10, and all are located in the M41-proteolytic domain of the protein. (iv) Bioinformatic analysis of the protein showed that the mutations alter the interaction of each monomer with the other five, symptomatic of an altered stability of the complex. Yeast complementation studies, beyond the scope of this investigation, would further establish the functional significance of these mutations.

The m-AAA mechanism of action is still under study, although recent investigations of its bacterial homologue FtsH indicate that the protease ring is stationary and that coordinated ATP-driven movements of each monomer accomplish substrate unfolding and translocation to the proteolytic chamber [START_REF] Augustin | An intersubunit signaling network coordinates ATP hydrolysis by m-AAA proteases[END_REF][START_REF] Bieniossek | The crystal structure of apo-FtsH reveals domain movements necessary for substrate unfolding and translocation[END_REF]. The electrostatic potential difference of the complex and the dipole of the central pore are altered in three of the six mutations that we detected. Furthermore, all six mutations diminish the interaction energy among monomers. These effects of the mutations suggest that each may have a negative effect on the m-AAA complex activity. For instance, the capacity of the complex to move substrates towards the proteolytic chamber through ATPase activity may be impaired by weaker interaction among monomers or by abnormal electrostatic potentials through the central pore [START_REF] Striebel | Controlled destruction: AAA+ ATPases in protein degradation from bacteria to eukaryotes[END_REF].

Five of the six AFG3L2 missense mutations previously reported, cluster in the proteolytic domain at amino acids Ser674, Glu691, Ala694, Glu700, and Arg702 (exon 16) [START_REF] Bella | Mutations in the mitochondrial protease gene AFG3L2 cause dominant hereditary ataxia SCA28[END_REF][START_REF] Edener | Early onset and slow progression of SCA28, a rare dominant ataxia in a large four-generation family with a novel AFG3L2 mutation[END_REF]. We found different mutations affecting the same amino acids (p.Gly671Arg and p.Gly671Glu; p.Met666Arg, p.Met666Thr, and p.Met666Val) indicating that exons 15 and 16 are mutational hotspots for SCA28, and consistent with the hypothesis that disruption of the peptidase domain is critical to the pathogenesis of SCA28.

Recent evidences from animal models also support the pathogenicity of AFG3L2 mutations: Afg3l2 knock-out mice presents with a severe neuromuscular phenotype, caused by a defect in motor axon and cerebellar development [START_REF] Maltecca | The mitochondrial protease AFG3L2 is essential for axonal development[END_REF], whereas the Afg3l2 Emv66 mouse, carrying a heterozygous loss-of-function mutation, develops a phenotype with similarities to SCA28 [START_REF] Maltecca | Haploinsufficiency of AFG3L2, the gene responsible for spinocerebellar ataxia type 28, causes mitochondriamediated Purkinje cell dark degeneration[END_REF]. Whole gene deletions were excluded in 300/366 patients and duplications in 129/366, suggesting that AFG3L2 deletions/duplications are not common in ADCAs. Moreover, two related patients with an 18p deletion encompassing AFG3L2 reportedly have not developed ataxia [START_REF] Nasir | Unbalanced whole arm translocation resulting in loss of 18p in dystonia[END_REF]. Copy number variants encompassing part or whole of AFG3L2 gene exist in normal individuals [START_REF] Kim | Identification of significant regional genetic variations using continuous CNV values in aCGH data[END_REF][START_REF] Redon | Global variation in copy number in the human genome[END_REF]. Therefore, in addition to the results of our screening designed to detect only large deletions and duplications, the absence of a phenotype in humans with duplications and deletions affecting AFG3L2, and the different phenotype observed in Afg3l2 null mice compared to Afg3l2 Emv66 mice, suggest that heterozygous deletions or duplications of the AFG3L2 gene may not cause SCA28. Looking at rearrangements in SCA28 should therefore not be prescribed in clinical practice.

Our survey shows that SCA28 is rare among European ADCA patients, accounting for ~3% of the 12 analyzed cases and ~1.5% of all ADCAs. As a comparison, polyglutamine expansions account for about 50%. The clinical and neuroradiological phenotype of the newly ascertained families is similar to that of the SCA28 patients reported so far [START_REF] Cagnoli | SCA28, a novel form of autosomal dominant cerebellar ataxia on chromosome 18p11.22-q11.2[END_REF][START_REF] Edener | Early onset and slow progression of SCA28, a rare dominant ataxia in a large four-generation family with a novel AFG3L2 mutation[END_REF][START_REF] Mariotti | Spinocerebellar ataxia type 28: a novel autosomal dominant cerebellar ataxia characterized by slow progression and ophthalmoparesis[END_REF]: disease onset is in adulthood and the disease progresses slowly with preserved functional capacity, decades after the diagnosis. The mean age at onset and age at onset variability is similar in our series and in the reported families: in five patients the onset was above 50 yrs, whereas two subjects were symptomatic at 6 and 8 yrs. Even in the young onset cases, the disease progressed relatively slowly, a much different pattern than has been observed in the polyglutamine diseases and evidence that SCA28 is not caused by cryptic polyglutamine expansion.

Interestingly, extrapyramidal features were not rare in our series of patients in contrast to previous reports, of potential value in speculating on the clinical diagnosis prior to genetic testing. As in many genetic diseases, particularly in those affecting the nervous system, the extent to which environmental factors affect disease phenotype, including onset age, selective neuronal involvement, and disease progression, remains to be determined.

Given the signs and symptoms of the disease, we suggest that the diagnosis of SCA28 should be considered in the presence of slowly progressive ataxia associated with oculomotor abnormalities. So far, AFG3L2 mutations seem to cluster in exons 10, 15, and 16, which should facilitate genetic diagnosis in clinical practice and a better understanding of the function and misfunction of AFG3L2.

Acknowledgments.

We are grateful to the family members for their participation, to the DNA and Cell bank of the Pedigrees of the nine families segregating a AFG3L2 mutation. Square and circles shadings: white = healthy; black = SCA28; gray = presumed SCA28; hash = asymptomatic subject carrying a disease haplotype. Five STR markers (on the left) were used to reconstruct the SCA28-surrounding haplotype that suggest a common ancestor for p.Thr541Ile, p.Met666Val and p.Gly671Arg mutations, in two families each. All subjects whose DNA was available (hyphen above the symbol) were tested for the AFG3L2 mutation segregating in the family.

F o r P e e r R e v i e w

Mutation screening.

The 17 coding exons of the AFG3L2 gene were PCR amplified in a total volume of 50 µl using 30 ng of genomic DNA, 144 µM dNTPs, 2 mM MgCl2, 1 U of Taq Gold (Applied Biosystems, Foster City, CA, USA) and 200 nM of forward and reverse primers reported in supplementary table S1.

The amplification of exon 1 required a final concentration of 1 M betaine (B0300, Sigma-Aldrich, St.Louis, MO, USA). Thermal cycling conditions were: 7 min at 95°C, followed by 14 cycles of 30 sec at 95°C, 30 sec at annealing temperature (Ta)+7°C-0.5°C/cycle (see supplementary table 1 and 1 min at 72°C; then 30 cycles consisting of X mn at 95°C, 30 sec at Ta (see supplementary table S1), and 1 min at 72°C. A final step of 40 cycles consisting of 1 min at 98°C-1°C/cycle was added in case of DHPLC analysis to increment heteroduplex formation.

The 17 coding exons of the SPG7 gene were amplified and directly sequenced. PCR was performed in a total volume of 25 µl, using 30 ng of genomic DNA, 144 µM dNTPs, 2.5 mM MgCl2, 0.5 U of Taq Gold (Applied Biosystems) and 200 nM of forward and reverse primers reported in supplementary table S2. The amplification of exon 1 required a final concentration of 1 M betaine.

Thermal cycling conditions for exons 1-13 were: 7 min at 95°C, followed by 14 cycles of 30 sec at 95°C, 30 sec at Ta+7°C-0.5°C/cycle (see supplementary table 2 and 1 min at 72°C; then 30 cycles consisting in X mn at 95°C, 30 sec at Ta, and 1 min at 72°C; with a final extension for 10 min at 72°C. Thermal cycling conditions for exons 14-17 were: 7 min at 95°C, 30 cycles at 95°C, 30 sec at Ta, and 1 min at 72°C; final extension for 10 min at 72°C. Fragments were purified using the ExoSAP method (MBI-Fermentas), and directly sequenced using the Big-Dye cycle sequencing kit and an ABI Prism 3100 Avant automatic sequencer (Applied Biosystems).

Minigene assay.

A minigene assay was set up to test the effect of AFG3L2 c.293-13_293-14delTT sequence variant on splicing. We amplified a 327 bp fragment from a subject carrying the variant. The forward The AAA ring does not form a regular hexagon and the contacts of one AAA domain with the two other interacting domains in the ring are much more asymmetric than those observed in the protease domain.

As a result, three different monomer conformations are present in 2CE7 structure.

In order to build the AFG3L2 hexamer model, we constructed, three different models based on the conformations of the three monomers present in 2CE7 structure, using the program NEST (http://trantor.bioc.columbia.edu/programs/jackal/index.html). Subsequently to obtain the hexamer model we superimposed them to the corresponding 2CE7 monomers in its hexamer using the program SHEBA [START_REF] Jung | Protein structure alignment using environmental profiles[END_REF]. Upon superimposition, the ADP molecule and the Mg2+ and Zn2+ ions were added to the model.

Using the program Yasara (www.yasara.org), we introduced the Thr654Ile, Met666Thr, Met666Arg, Met666Val, Gly671Glu and Gly671Arg mutations. Next we refined the model using the program Yasara with the protocol described below.

Each model was refined performing a 150 ps simulation of the homology models using the protocol described in [START_REF] Krieger | Making optimal use of empirical energy functions: force-field parameterization in crystal space[END_REF]. It saves one PDB file every 25 picoseconds, and allows the user to identify the best snapshot giving as output a table with force field energies, PhiPsi, Backbone and Packing3 WHATIF checks [START_REF] Vriend | WHAT IF: a molecular modeling and drug design program[END_REF]. The conformer having the lowest value of Q was selected for subsequent analyses.

Data on the electric dipole moment calculated using the Yasara software. The difference of electrostatic potential (mV) was calculated from the measure of the electric dipole moment using the formula:

( )

0 2 4π ε r d = ∆V ⋅ ⋅
. D = electric dipole moment; r = distance between examer surfaces; ε0 = electric constant in vacuum.

The pH of the AFG3L2 hexamer catalitic site was calculated using pH Calculator software (http://www.webqc.org/phsolver.php). pKa values were obtained from the PROPKA software ver 2.0 [START_REF] Guerois | Predicting changes in the stability of proteins and protein complexes: a study of more than 1000 mutations[END_REF]. We calculated the pH of normal hexamer and hexamers containing identified mutations for all aminoacids distant <0.5 nm from the Zn atom within the catalytic site (constant arbitrary concentration set at 0.01M).

The energetic stability of AFG3L2 protein and the interaction energy of each monomer were calculated using the FoldX software ver 3.0 [START_REF] Guerois | Predicting changes in the stability of proteins and protein complexes: a study of more than 1000 mutations[END_REF]. We calculated both the interaction energy between one monomer and the two flanking monomers, and between one monomer and the remaining hexamer.

To verify the possible three-dimensional structure of AFG3L2 homopolymer or heteropolymer, a 3D model of the C-terminal tract of SPG7, a known interactor of AFG3L2 [START_REF] Atorino | Loss of m-AAA protease in mitochondria causes complex I deficiency and increased sensitivity to oxidative stress in hereditary spastic paraplegia[END_REF] was also created following the same protocol as above.

Homo-and Heteropolymers were reconstructed with the software NEST and SHEBA [START_REF] Jung | Protein structure alignment using environmental profiles[END_REF][START_REF] Petrey | Using multiple structure alignments, fast model building, and energetic analysis in fold recognition and homology modeling[END_REF].

Finally, the software PoPMuSic estimated the stabilizing/destabilizing effect of each aminoacidic substitution, through the measurement of the free energy difference (∆∆G) of the molecule [START_REF] Gilis | PoPMuSiC, an algorithm for predicting protein mutant stability changes: application to prion proteins[END_REF]. In bold, variants not found among controls and/or in the dbSNP(129). Controls were all of French origin, except for note 3 -95 Italian controls, and note 6 -95 Italian and 95 French controls; 1-The effect on splicing was tested using a minigene assay (see Supp. Materials and Methods); 2-The c.400-14C>G variant was tested in one family and did not segregate from the affected father to the affected son; 4-Two hundred-24 cases were homozygous (A/A) and 125 heterozygous (G/A); n.a. = not available; 5-Fifty-seven controls were homozygous (A/A) and 28 heterozygous (G/A). S4. Mutation effect on the electric dipole of the translocation channel and on the potential difference between the inner mitochondrial membrane side and the matrix side of the hexamer. -51.46 -70.63 -59.66 -38.88 -64.79 -50.73 -56.03

F

G671R

1 Yellow = wild-type (wt) interaction; Green = lower interaction; Blue = slightly lower interaction § Interaction energy was calculated using the application FoldX ver3.0 [START_REF] Guerois | Predicting changes in the stability of proteins and protein complexes: a study of more than 1000 mutations[END_REF]

  of AFG3L2 mutations and the clinical features of this disease.

  ), and directly sequenced using the Big-Dye terminator cycle sequencing kit ver.

  Centre de Recherche de l'Institut du Cerveau et de la Moelle épinière for technical assistance and to Drs C Marelli, J Yaouang, M Ponsot, H Chneiweiss, D Ménard, S Schaeffer, A Lagueny, L Carluer and Y Boukhriche as well as Prof A Harding for referring some of the patients and for clinical examinations.

  FIGURE LEGENDS

  Figure 2.

Figure 3 .

 3 Figure 3.

  EcoRIrestriction site, and the reverse primer (5'-TACGTCTAGAAATGCCTCCCAACCTTCTCT) introduced a XbaI restriction site. The fragment was sequenced to exclude the presence of PCRintroduced mutations. The PCR product was cloned into a pTZ57R/T vector using the InsTA Cloning kit (Invitrogen), and transformed in DH5α bacterial cells following manufacturer's recommendations (RBC Bioscience, Chung Ho City, Taiwan). Plasmids were extracted using standard methods, checked by sequencing, and digested with EcoRI and XbaI restriction endonucleases. The insert was gel-purified and sub-cloned into a pAltermax modified vector (kindly provided by dr. Gareth Eldvige, Wellcome Trust Centre, Oxford, UK, and dr. Ivana Kurelac, University of Bologna). Plasmids containing the wild-type or the variant sequence were used to transform HEK293 cells using the TurboFect kit (Fermentas). After 48h, RNA was extracted with the RNeasy Kit (Qiagen), retrotranscribed with the GoScript kit (Promega). The cDNA was amplified and sequenced with vector primers.To build a three dimensional model of the C-terminal region of AFG3L2, the sequence corresponding to residues 306-762 was submitted to PDB-blast (http://protein.cribi.unipd.it/pdbblast/) that recognized the cell division bacterial protein FtsH with an R-score of 1E-117. Its crystal structure was present in the protein database (PDB) as 2CE7, which contains the AAA domain and a protease domain. Crystallization experiments of 2CE7 gave rise to a tetragonal crystal form containing six monomers per asymmetric unit. These belong to two virtually identical half-hexamers; the complete hexameric molecules are generated by a crystallographic two-fold axis.

  choose the best conformer saved during the refinement procedure we derived a quality parameter defined as follows: Q = (Energy/10000) -(PhiPsi+Backbone+Packing3) = quality

  .,

	SCA28 Cagnoli C
	F o r
	P
	e
	e r
	R
	e
	v i e
	w

  Table 1 reports the clinical and neuroradiological features of twenty-five SCA28 patients from ten

	SCA28 Cagnoli C
	F o r
	P
	e
	e r
	R
	e
	v i e
	w

Table 1 .

 1 Clinical features of SCA28 patients.

	Cerebellar Signs

Notes: LL, lower limbs; UL, upper limbs; N, normal; +, augmented; NA, not available;

Table S3 .

 S3 AFG3L2 variants found in 366 ADCA patients.

	Supp. Variants		Exon / Intron	Patients	Controls	Notes
	rs12327346:C>G		EX1	20/366	7/95
	c.-92T>C			1/366	0/95
	c.-71C>T			4/366	1/95
	c.-48G>A			2/366	0/95
	c.-32C>T			1/366	0/95
	c.293-33G>A		IVS3	1/366	0/95	The variant does not alter splicing
	c.293-13_293-14delTT			2/366	0/95	The variant does not alter splicing 1
	c.400-14C>G		IVS4	4/366 2	0/95	The variant does not alter splicing
	c.718C>A (p.=)		EX7	1/366	n.a.	The variant does not alter splicing
	rs8097342:G>A rs8091858:C>T c.1027-12T>C rs9966470:A>T	IVS7 IVS8 r o IVS9	120/366 27/366 2/366 85/366	n.a. 0/95 0/95 3 21/95	The variant does not alter splicing
	c.1165-24G>A c.1319-59G>T c.1319-40C>T rs11080572:G>A c.1426+21_1426+22insCAGGTC c.1479G>A (p.=)	5/366 3/366 1/366 349/366 4 e IVS10 P e EX11 IVS11 1/366 r EX12 1/366	1/95 0/95 0/95 85/95 5 0/95 0/95	The variants does not alter splicing The variants does not alter splicing The variant does not alter splicing The variant does not alter splicing
	rs11553521:A>G		EX13	R e 125/366 26/95
	c.1664-39G>A c.1664-9T>C c.2175+18G>A		IVS13 IVS16	8/366 1/366 9/366	2/95 v 0/95 i e 4/190 6	The variant does not alter splicing
	rs1129115:G>C c.2325C>T (p.=)		EX17	117/366 1/293	26/95 w 0/95 Not found in an affected niece
				7

  -type (wt) values; Green = lower dipole and potential difference; Blue = dipole and potential difference not significantly different from the wild-type; Red = higher dipole and potential difference. mV = millivolts; C m = Coulomb meter.Supp. TableS5. Mean interaction energy § (Kcal/Mol) between each monomer of the AAAprotease (A to F) and the other five monomers

		Potential difference -PD	PD compared to wild	PD compared to wild	Electric dipole
			(mV)		type (mV)		type (%)	(C m)
	wt		22,274					1,79*10 -26
	T654I		22,60		0,33		1,48%	1,82*10 -26
	M666V		22,305		0,031		0,14%	1,79*10 -26
	M666T		22,945		0,671		3,01%	1,85*10 -26
	M666R		19,548		-2,726		-12,24%	1,57*10 -26
	G671E		25,736		3,462		15,54%	2,07*10 -26
	G671R		18,887		-3,387		-15,21%	1,52*10 -26
	Yellow = wildMonomer
		A	B	C	D	E	F	Mean 1
	wt	-49.89	-79.31	-57.50	-59.64	-82.24	-54.58	-63.86
	T654I	-43.23	-76.66	-66.77	-55.32	-76.86	-57.56	-62.73
	M666V	-56.73	-83.46	-51.03	-37.19	-60.82	-56.99	-57.70
	M666T	-50.50	-64.60	-46.42	-48.36	-76.81	-58.26	-57.49
	M666R	-35.73	-71.15	-49.36	-37.83	-58.24	-42.46	-49.13
	G671E	-49.41	-79.02	-54.77	-61.61	-77.72	-50.24	-62.13

John Wiley & Sons, Inc.Human Mutation

Supp. Table S1