Alexandre Seuret

Christophe Prieur

Christophe Prieur Event

Event-triggered sampling algorithms based on a Lyapunov function

come

Event-triggered sampling algorithms based on a Lyapunov function

Alexandre Seuret and Christophe Prieur

Abstract-Usually feedback laws are implemented in a periodic fashion on digital hardware. The main reason for using this periodicity in the hardware comes from the difficulties to analyze the stability of aperiodic or asynchronous systems. However it also seems natural to hold the same control input longer if the system behaves in a suitable way or shorter if the system requires an updated input. In this paper, an algorithm is suggested to sample the control input based on the behavior of a Lyapunov-like function. This algorithm is called eventtriggered since the Lyapunov-like function directly depends on the state of the systems.

I. INTRODUCTION

Over the years, research in control of nonlinear dynamical systems has lead to many different tools to design globally asymptotically stabilizing feedbacks, see e.g. [START_REF] Byrnes | New results and examples in nonlinear feedback stabilization[END_REF], [START_REF] Freeman | Robust nonlinear control design. State-space and Lyapunov techniques[END_REF], [START_REF] Khalil | Nonlinear Systems[END_REF], [START_REF] Kokotovic | The joy of feedback: nonlinear and adaptative[END_REF]. In the quest for providing more flexible tools for achieving the stabilization and performance tasks, research efforts have focused on analyzing the effect of the switching or on-line adaptation, in particular when implementing the controllers in digital plateforms (see [START_REF] Anta | To sample or not to sample: self-triggered control for nonlinear systems[END_REF] and references therein). This may complicate the analysis of the stability, but also may reduce the cost, i.e. the number of times that the feedback control uses the microprocessors, when closing the loop in digital platforms. Traditionally, the sampling for the controller is chosen periodic, since the analysis of the discrete-time systems has been widely investigated for linear systems [START_REF] Chen | Optimal sampled-data control systems[END_REF]. Then several studies deal with the robustness of sampled-data controllers with respect to uncertainties in the sampling instants sequence [START_REF] Fridman | Robust sampled-data stabilization of linear systems: An input delay approach[END_REF], [START_REF] Fujioka | Stability analysis of systems with aperiodic sample-andhold devices[END_REF], [START_REF] Seuret | A novel stability analysis of sampled-data systems with applications to multi-rate sampling and packet loss[END_REF]. These methods ensure the stability of a linear sampled-data system if the sampling period is included in a certain interval. These technics are very relevant but they consider the worst situation. In recent years, an interesting method so-called Event-Based Control suggests to adapt the sampling sequence to some events concerning the state of the system [START_REF] Ȧström | Comparison of Riemann and Lebesgue sampling for first order stochastic systems[END_REF], [START_REF] Heemels | Analysis of eventdriven controller for linear systems[END_REF], [START_REF] Sandee | Event-driven control as an opportunity in the multidisciplinary development of embedded controller[END_REF], [START_REF] Velasco | On Lyapunov sampling for eventdriven controllers[END_REF].

One application of this kind of algorithms is for the Networked Control Systems (see [START_REF] Hespanha | A survey of recent results in networked control systems[END_REF], [START_REF] Zampieri | A survey of recent results in Networked Control Systems[END_REF]). Such systems are controlled systems containing several distributed plants which are connected through a communication network. In this situation, the controlled system works in continuoustime whereas the controller provides an discrete-time input which is hold during a sampling period. In this situation, it is relevant to reduce the among of control updates as suggested in [START_REF] Ȧström | Comparison of Riemann and Lebesgue sampling for first order stochastic systems[END_REF].

Alexandre Seuret and Christophe Prieur are with Department of Automatic Control, Gipsa-lab, 961 rue de la Houille Blanche, BP 46, 38402 Grenoble Cedex, France, alexandre.seuret@gipsa-lab.grenoble-inp.fr, christophe.prieur@gipsa-lab.grenoble-inp.fr

To tackle this issue, the problem of the design of an eventtriggered algorithm is first rewritten in this paper as the stability study of a system with a mixed continuous/discrete dynamics (also called hybrid system), as considered e.g. [START_REF] Goebel | Hybrid dynamical systems[END_REF], [START_REF] Prieur | Hybrid feedback control and robust stabilization of nonlinear systems[END_REF], [START_REF] Prieur | Guaranteed stability for nonlinear systems by means of a hybrid loop[END_REF] in a different context. Using this framework and the Lyapunov theory that is now well known on this kind of nonlinear systems, we compute two new event-triggered algorithms for the implementation of feedback controllers. The first event-triggered algorithm makes a Lyapunov-like function decrease (see Theorem 1 below for a precise statement), whereas the second one allows, in general, to sample less times the control loop to ensure the asymptotic stability of the closed-loop system (see Theorem 2 below). These results are first derived for nonlinear control systems for which it is known a (nonlinear) stabilizing controller. Then the method is particularized to linear control systems and illustrated on (nonlinear and linear) systems borrowed from the literature.

The paper is organized as follows. In Section II some materials on hybrid systems are given, and the problem under consideration in this paper is introduced. In Section III, two event-triggered algorithms are given for nonlinear control systems. Then both main results are applied to the linear case in Section IV. Some numerical simulations illustrate the stability results and the performances of both algorithms in the nonlinear and in the linear cases (see Section V). Section VI contains some concluding remarks and points out some possible open research lines.

Notation. Throughout the article, the sets N, R + , R n , R n×n and S n denote respectively the sets of positive integers, positive scalars, n-dimensional vectors, n × n matrices and symmetric matrices of R n×n . The notation | • | stands for the Euclidean norm. Given a compact set A, the notation |x| A = min{|x -y|, y ∈ A]} indicates the distance of the vector x to the set A. The superscript 'T ' stands for matrix transposition. A function µ is said to be of class K ∞ if it is continuous, zero at zero, increasing and unbounded. The symbols I and 0 represent the identity and the zero matrices of appropriate dimensions.

II. PROBLEM FORMULATION

Consider a continuous-time nonlinear system

ẋ = f (x, u), x(t 0 ) = φ 0 , (1)
where x ∈ R n and u ∈ R m stand respectively for the state variable and the input vector, φ 0 ∈ R n is the initial state and

f : R n × R m → R n is a locally Lipschitz function.
Assume that the system (1) is globally asymptotically stabilizable, i.e. that there exist a Lyapunov function V and a state feedback control law ū such that the derivative of the Lyapunov function along the trajectories of the closed-loop system is negative definite. This means that:

Assumption 1: There exist a continuously differentiable function V : R n → R, some functions µ 1 , µ 2 and µ 3 in K ∞ and a continuous controller u : R n → R m such that u(0) = 0 and for all x ∈ R n ,

µ 1 (|x|) ≤ V (x) ≤ µ 2 (|x|), ∇V (x).f (x, u(x)) ≤ -µ 3 (|x|).
The control law u has been designed in continuoustime. This means that the (continuous)-time derivative of a Lyapunov function is negative definite.

In practice, it is not realistic to implement a control law in continuous-time. As the control input is computed in a digital hardware, only a sampled version of the input is implemented in the actuators. Generally, the sampling is chosen periodic and with a small period so that the sampled signal is very close to the continuous one. However the computation of the control values is not done instantaneously. It requires a minimum sampling period which guarantees that the controller is able to compute the correct data on time. Consequently, the use of a small sampling period requires an efficient processor allowing to compute the control value in short time. An alternative solution is to develop an algorithm which triggers the sampling period with respect to the state of the system, as shown in Figure 1. The contribution of this paper is to let the system decide by itself if an update of the control is needed or not. In order to clarify the notation and the problem under consideration, a hybrid formulation of the sampled-data system is proposed. It was indeed shown in [START_REF] Goebel | Hybrid dynamical systems[END_REF], [START_REF] Prieur | Hybrid feedback control and robust stabilization of nonlinear systems[END_REF], [START_REF] Prieur | Guaranteed stability for nonlinear systems by means of a hybrid loop[END_REF] that the sampled-data system can be rewritten as

   ẋ = f (x, s), ṡ = 0, ṗ = g(x, s, p), if (x, s, p) ∈ F ,    x + = x, s + = u(x), p + = g 0 (x, s, p), if (x, s, p) ∈ J , (2)
where s ∈ R m represents the held value of the control input, p ∈ R q contains additional parameters, g : R n × R m × R q → R q and g 0 : R n × R m × R q → R q are two continuous functions of appropriate dimensions, and F and J are two subsets of R n × R m × R q . These sets are respectively called flow set and jump set and are degrees of freedom of the event-triggered algorithm. The variable s represents the held value of u(x), which is the control value that is implemented over the sampling interval.

We recall some basic ingredients on hybrid system theory, and on the notion of solutions of (2) (see [START_REF] Goebel | Hybrid dynamical systems[END_REF], [START_REF] Prieur | Hybrid feedback control and robust stabilization of nonlinear systems[END_REF]). Due to mixed discrete/continuous dynamics, a solution of (2) will be defined on a mixed discrete/continuous time domain. More precisely, a set E is a hybrid time domain if for all (T, J) ∈ E, E ∩ ([0, T ] × {0, 1, . . . J}) 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 (2) consists of a hybrid time domain dom and functions x : dom → R n , s : dom → R m , and p : dom → R such that (x, s, p)(t, j) is absolutely continuous in t for a fixed j and (t, j) ∈ dom satisfying (S1) for all j ∈ N and almost all t such that (t, j) ∈ dom,

x(t, j) ∈ F, ẋ(t, j) = f (x(t, j), s(t, j)), ṡ(t, j) = 0, ṗ(t, j) = g(x(t, j), s(t, j), p(t, j)),
(S2) for all (t, j) ∈ dom such that (t, j + 1) ∈ dom,

x(t, j) ∈ J, x(t, j + 1) = x(t, j), s(t, j + 1) = u(x(t, j)), p(t, j + 1) = g(x(t, j), s(t, j), p(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 ( 2) is said to be complete if dom is unbounded, Zeno if it is complete but the projection of dom onto R ≥0 is bounded, and maximal if there does not exist an other solution x of (2) such that x is a truncation of x to some proper subset of dom. Hereafter, only maximal solutions will be considered. For more details about this hybrid systems framework, we refer the reader to [START_REF] Goebel | Hybrid dynamical systems[END_REF], [START_REF] Prieur | Hybrid feedback control and robust stabilization of nonlinear systems[END_REF]. The following definition describes the requirements to prove the global asymptotic stability of the solutions of (2).

Definition 2.1:

Given a compact subset A of R n × R m × R q the hybrid system (2) is said to be • stable to A: if for each ε > 0 there ex- ists δ > 0 such that each solution (x, s, p) to (2) with |(x(0, 0), s(0, 0), p(0, 0))| A ≤ δ satisfies |(x(t, j), s(t, j), p(t, j))| A ≤ ε for all (t, j) ∈ dom; • attractive to A: if every solution x to (2) is complete and satisfies lim t+j→∞ |(x(t, j), s(t, j), p(t, j))| A = 0;
• globally asymptotically stable to A: if it is both stable and attractive to A. Given an initial condition (φ 0 , s 0 , p 0 ) in R n ×R m ×R, and a solution (x, s, p) of (2) defined on a hybrid time domain dom, the set of the sampling time instants, when the control input is updated, (plus 0) is denoted T and is {t j }, where t j is such dom = ∪ j∈J ([t j , t j+1 ], j). Among other results, we will prove in this paper some properties on the set T depending on the choice of the event-triggered algorithm. In particular, in our hybrid systems framework, T is at most countable.

In this paper, different sets F and J are defined, and thus different event-triggered algorithms are considered. Let, as a first example, the particular case where p = τ ∈ R and such that the dynamics of the system is rewritten, for any

T > 0 as    ẋ = f (x, s), ṡ = 0, τ = 1, if (x, s, τ ) ∈ F T ,    x + = x, s + = u(x), τ + = 0, if (x, s, τ ) ∈ J T , (3)
where F T and J T are the following subsets of

R n × R m × [0, T ]: F T = {x, s, τ, such that τ ≤ T }, J T = {x, s, τ, such that τ ≥ T }. ( 4
)
As shown in [START_REF] Goebel | Hybrid dynamical systems[END_REF], the hybrid model expresses the case of periodic sampling. In this simple algorithm, after each jump, the solution is either at the equilibrium or has to flow. It avoids the existence of Zeno solutions, and also it reduces the complexity when implementing the eventtriggered algorithm. Of course, in general, the system (3) is not globally asymptotically stable since the update of the control law does not depend on the system position but is done periodically. This motivates us to consider the following problem: Problem 1: Define appropriate sets F and J and dynamics of the variable p such that, after each jump of the solutions of ( 2), the solutions have to flow, and such that ( 2) is globally asymptotically stable.

III. EVENT-TRIGGERED ALGORITHM FOR NONLINEAR SYSTEMS

A. A continuous-time decreasing function

In this section, the objective is to define flow and jump sets which are based on the decay of the function in continuoustime.

Theorem 1: Under Assumption 1, consider a given function µ of class K ∞ such that µ(r) < µ 3 (r), for all r > 0.

Consider the flow and jump sets given by

F 1 = {(x, s), ∇V (x).f (x, s) ≤ -µ(|x|)}, J 1 = {(x, s), ∇V (x).f (x, s) ≥ -µ(|x|)}, ( 5
)
and the associated event-triggered algorithm. Then the system ( 2) with F = F 1 and J = J 1 (and without state p) is globally asymptotically stable to the origin. Moreover, after each jump of the solutions of this hybrid system, either the state is the origin or the solution has to flow.

Proof. The proof of Theorem 1 is based on the decreasing property of the function V given by Assumption 1, along the solutions of (2), with F and J given by [START_REF] Freeman | Robust nonlinear control design. State-space and Lyapunov techniques[END_REF]. See [START_REF] Prieur | Guaranteed stability for nonlinear systems by means of a hybrid loop[END_REF] for analogous ideas for a different problem.

Given a switching time instant t 0 ∈ T , denoting (with a slight abuse of notation) x(t + 0 ) the state after the jump (and similarly for the other variables), using Assumption 1, it yields

∇V (x(t + 0 )).f (x(t + 0 ), s(t + 0 )) = ∇V (x).f (x(t + 0 ), u(x(t + 0 ))) ≤ -µ 3 (|x(t + 0 )|) ≤ -µ(|x(t + 0 )|) -ε(|x(t + 0 )|), where ε(|x(t + 0 )|) = µ 3 (|x(t + 0 )|)-µ(|x(t + 0 )|
) is non-negative and equals 0 only if x(t + 0 ) is vanishing. Thus, after a jump, two cases may occur: 1) either the state x is at the origin (and the same for the other components), and then the solution remains at the origin; 2) or (x(t + 0 ), s(t + 0 )) belongs to F 1 and is different to the origin.

Consider now (x, s) in F 1 \ {0}. Then we get

∇V (x).f (x, s) = ∇V (x).(f (x, s) -f (x, u(x))) +∇V (x).f (x, u(x)),
and using Assumption 1, we obtain

∇V (x).f (x, s) ≤ -µ 3 (|x|) +∇V (x).(f (x, s) -f (x, u(x))).
Then, the solution (x, s) of system (2) with F = F 1 and J = J 1 stays in F 1 until a state x = x * (if such a state does exist) defined by

∇V (x * ).(f (x * , s) -f (x * , u(x * ))) = µ 3 (|x * |) -µ(|x * |).
Two subcases may occur. 2.a) If there exists such x * , then the couple (x * , s) belongs to J 1 , and by definition of s + , (x * + , s + ) belongs to F 1 . 2.b) If there does not exist such x * , then the solution of the system (2) stays in F 1 .

For both cases, the derivative of V is negative while (x, s) is in F 1 and V is constant while (x, s) is in J 1 . This implies that the system (2) with F = F 1 and J = J 1 is stable.

To prove the attractivity of the system (2) with F = F 1 and J = J 1 , let us apply the LaSalle invariance property for hybrid systems. Let us consider a solution of this hybrid system which is included in a level set of the function V . Let us show that this solution should be equal to 0.

The solution cannot jump (since after a jump, the solution has to flow, and thus the value of V has to decrease). Given a solution flowing for all time, due to Assumption 1, the state x cannot stay at the level set of V . Thus the solution has to be constant and equal to the origin. Therefore, by [9, Theorem 23, page 64], the system (2) with F = F 1 and J = J 1 is globally asymptotically stable. This concludes the proof of Theorem 1.

• Remark 1: A main improvement of the proposed method compared for example to [START_REF] Anta | To sample or not to sample: self-triggered control for nonlinear systems[END_REF], is that no Input-to-State Stability (ISS) assumption for system (1) is needed. More precisely the method that is suggested in [START_REF] Anta | To sample or not to sample: self-triggered control for nonlinear systems[END_REF] requires the existence of functions α and γ of class K ∞ , such that, for all x in R n ,

∇V (x).f (x, s) ≤ -α(|x|) + γ(|u(x) -s|) .
Then the event-triggered algorithm is defined by a condition on the error between the control value s and the controller u that is:

|u(x) -s| ≤ γ -1 (σα(|x|)),
where σ in (0, 1) is a tuning parameter. In the present article, instead of an ISS assumption, only the global asymptotic stability is needed. As remarked in [START_REF] Sontag | Input to state stability: Basic concepts and results[END_REF] e.g., Assumption 1 is weaker than the ISS property, and it is sufficient to define the event-triggered algorithm by the value of the derivative of the Lyapunov function along the trajectories of the system.

• Remark 2: Another important issue concerns the possibility that the solution of the system for a given initial condition, never reaches the set J 1 . In such situation, it simply means that the system is already asymptotically stable without any control (or with a constant control value) and the control law does not need to be updated. This situation is not taken into account in the method proposed in [START_REF] Anta | To sample or not to sample: self-triggered control for nonlinear systems[END_REF]. • Remark 3: On the other side, the main drawback of the method proposed in this article is the real time computation of the derivative of the Lyapunov function V .

• Note that picking µ = 0 in Theorem 1 gives a partial result and allows to design an event-triggered algorithm such that the closed-loop system is globally stable. More precisely we have Proposition 1: Under Assumption 1, consider the flow and jump sets given by

F 1 = {(x, s), ∇V (x).f (x, s) ≤ 0}, J 1 = {(x, s), ∇V (x).f (x, s) ≥ 0},
and the associated event-triggered algorithm. Then the systems [START_REF] Ȧström | Comparison of Riemann and Lebesgue sampling for first order stochastic systems[END_REF] with F = F 1 and J = J 1 is globally stable. Moreover, after each jump of the solutions of this hybrid system, either the state is the origin or the solution has to flow.

Proof. The proof follows the lines of Theorem 1. More precisely, we may check that by selecting F = F 1 and J = J 1 , and by using Assumption 1, the derivative of the Lyapunov function V is negative while the state of the solution (x, s) of ( 2) is in F 1 and is constant (x, s) is in J 1 . This implies, with [9, Theorem 23, page 64], that the system (2) with F = F 1 and J = J 1 is stable.

Finally, using Assumption 1 again, we note that, given a solution of (2) with F = F 1 and J = J 1 , after each jump (if such a jump does exist), either the state is the origin or the solution has to flow.

•

B. A discrete-time decreasing function

In this section, the objective is to define flow and jump sets which are based both on the decay of the function in continuous-time and in discrete-time. A second eventtriggered algorithm is given by the following system

       ẋ = f (x, s), ṡ = 0, v = 0, τ = 1, if (x, s, v, τ ) ∈ F,        x + = x, s + = u(x), v + = λV (x), τ + = 0, if (x, s, v, τ ) ∈ J , ( 6
)
where v is an additional state in R, λ ∈ (0, 1) . This is system (2) with a particular choice of the function g and g 0 . A design of an event-triggered algorithm for the control system (1) is given by the following: Theorem 2: Under Assumption 1, consider a sufficiently large T > 0 and a function µ of class K ∞ such that µ(r) < µ 3 (r), for all r > 0. Consider the flow and jump sets given by

F 2 = F1 F2 F T , J 2 = J1 J2 J T , where F1 = {(x, s, v, τ ), (x, s) ∈ F 1 } , J1 = {(x, s, v, τ ), (x, s) ∈ J 1 } , F2 = {(x, s, v, τ ), V (x) ≤ v} , J2 = {(x, s, v, τ ), V (x) ≥ v} ,
and where F T and J T are defined by (4). Then the system (6) with the event-triggered algorithm derived from F = F 2 and J = J 2 is globally asymptotically stable to the set {0} × {0} × {0} × [0, T ]. Moreover, after each jump of the solutions of this hybrid system, either the state is the origin or the solution has to flow.

Proof. The introduction of the sets F T and J T ensures that there are at most T units of continuous time between two jumps. Moreover, for each solution of the system (6) with the event-triggered algorithm derived from F = F 2 and J = J 2 , due to the discrete dynamics equation, after each jump, the solution has to flow.

Note that since the flow set F 2 and the jump sets J 2 are closed, and since the dynamics of the system (6) are continuous, then, with [9, Thereom S3], each maximal solution is either complete (i.e. with a unbounded time domain) or blows up in finite time.

Let us show, by stating a contradiction, that the solution cannot blow up in finite time. To do that let us assume that it is given a solution of ( 6) with the event-triggered algorithm derived from F = F 2 and J = J 2 such that the solution blows up in (hybrid) finite time. Inspecting the variables x, s, v and τ , if the solution is blowing up in finite continuous time, then the x-variable has to blow up in finite continuous time (the other components are bounded). However, by definition of F 1 , the V function cannot blow up. Therefore, if there exists a blow up in (hybrid) finite time, then it should blow up in the discrete time direction. Now let us remark that between two jumps, due the dynamics of the v variable, the V function cannot increase. It gives a contradiction with the blow up of the x-variable in discrete time.

Let us state now that the system (6) with the eventtriggered algorithm derived from F = F 2 and J = J 2 is globally attractive to {0}×{0}×{0}×[0, T ]. To do that, let us consider a solution (x, s, v, τ ) of system [START_REF] Fridman | A refined input delay approach to sampled-data control[END_REF] with the eventtriggered algorithm derived from F = F 2 and J = J 2 . Moreover let us denote T the set of (continuous) sampling time instants, when the control input is updated (plus 0).

The set T = {t k } is infinite since the solution is complete, and, between two jumps, there is at most T units of time. Let us denoting V k , and x k the value of the V function and of the x-variable, when there is a jump. Due to Assumption 1 and the inequality µ(r) < µ 3 (r) for all r > 0, we have V k < V k-1 , for all k in N, and thus x k converges to zero when k tends to the infinity. Now, by the continuous differentiability of the functions in the right-hand side of the system (6), we have the convergence of the s-and v-components of the solution at the switching time instants. Thus, by definition of the set F2 , and since the v-variable converges to zero at the switching time instants, the x-variable tends also to zero when the hybrid time tends to the infinity. By the continuous differentiability of the functions in the right-hand side of the system (6), it concludes the proof of the attractivity of the system (6) with the event-triggered algorithm derived from F = F 2 and J = J 2 to the set {0} × {0} × {0} × [0, T ].

The local stability of the system (6) with the eventtriggered algorithm derived from F = F 2 and J = J 2 follows from the fact that between two switches the V function does not increase and that the V function is bounded by the v-variable between two switches.

This concludes the proof of Theorem 2.

• Remark 4: Compared to the event-driven algorithm which is described in Theorem 1, the above one could lead to a larger sampling period since it allows the function V to be locally increasing. This is the main improvement of the second algorithm with respect to the first one (as remarked on the numerical simulations of Section V below). On this other hand the performance and the convergence rate of the system might be worst than using Theorem 1 since the second algorithm allows an increase of the function V . Here appears a tradeoff between the minimization of the number of updates and the performance of the closed-loop system. We will come back to this issue in the conclusion section. • Remark 5: An important issue has to do with the choice of the parameter λ. If λ is chosen too small, it may happen that the solutions of the system never reaches F2 . This means that the event-driven algorithm defined by the sets F 2 and J 2 is exactly the same than the one defined in Theorem 1. Another interest of the event-driven algorithm which is proposed in Theorem 2 is that there is no need to ensure the existence of a particular value of λ such that the sets F 2 is not empty.

•

IV. APPLICATION TO LINEAR SYSTEMS A. A continuous-time decreasing function

Consider now a linear system of the form

ẋ = Ax + Bu, (7)
where x ∈ R n and u ∈ R m stand for the state variable and the input vector. The matrices A and B are constant and known and of appropriate dimensions. Let us assume that the pair (A, B) is controllable. Then the proposed control law for this system is u = Kx where K in R m×n is such that the matrix A + BK is Hurwitz. There also exist a positive scalar α and a symmetric positive definite matrix P so that

P (A + BK) + (A + BK) T P < -2αP. ( 8
)
Thus Assumption 1 holds with V (x) = x T P x and u(x) = Kx for all x ∈ R n . Rewriting the closed loop system in a hybrid framework, we get:

ẋ = Ax + Bs, ṡ = 0, if (x, s) ∈ F, x + = x, s + = Kx, if (x, s) ∈ J . ( 9
)
By noting that

∇V (x)f (x, s) = x s Π 1 x s
where

Π 1 = P A + A T P + 2 ᾱP P BK (BK) T P 0 , ᾱ ∈ (0, α),
the following result follows readily from Theorem 1: Proposition 2: Assume there exist a symmetric positive definite matrix P in R n×n , a matrix K in R n×m and a positive scalar α satisfying [START_REF] Fujioka | Stability analysis of systems with aperiodic sample-andhold devices[END_REF]. Consider ᾱ ∈ (0, α) and the flow and jump sets defined by

F 1L = (x, s), x s T Π 1 x s ≤ 0 , J 1L = (x, s), x s T Π 1 x s ≥ 0 .
Then the system (9) with the event-triggered algorithm derived from F = F 1L and J = J 1L is globally asymptotically stable to the origin. Moreover, after each jump of the solutions of this hybrid system, either the state is the origin or the solution has to flow.

B. A discrete-time decreasing function

Following the idea proposed in Theorem 2, we will consider the hybrid system defined by

       ẋ = Ax + Bs, ṡ = 0, v = 0, τ = 1, if (x, s, v, τ ) ∈ F,        x + = x, s + = Kx, v + = λV (x), τ + = 0, if (x, s, v, τ ) ∈ J . ( 10
)
By applying Theorem 2, we obtain the following: Proposition 3: Assume there exist a symmetric positive definite matrix P in R n×n , a matrix K in R n×m and a positive scalar α satisfying [START_REF] Fujioka | Stability analysis of systems with aperiodic sample-andhold devices[END_REF]. Consider ᾱ ∈ (0, α) and the flow and jump sets given by

F 2L = F1L F2L F T , J 2L = J1L F2L F T , where F1L = {(x, s, v, τ ), (x, s) ∈ F 1L } , J1L = {(x, s, v, τ ), (x, s) ∈ J 1L } , F2L = (x, s, v, τ ), x T P x ≤ v , J2L = (x, s, v, τ ), x T P x ≥ v .
Then the system [START_REF] Heemels | Analysis of eventdriven controller for linear systems[END_REF] with the event-triggered algorithm derived from F = F 2L and J = J 2L is globally asymptotically stable to the set {0} × {0} × {0} × [0, T ]. Moreover, after each jump of the solutions of this hybrid system, either the state is the origin or the solution has to flow.

C. Comments on the linear case

The event-triggered algorithms which are exposed above do not provide any information of the duration while a control law is held. In the sequel, a complementary analysis is provided for the case of linear systems to give an upperbound and a lower bound of the holding times.

Let χ ∈ R n be the value of x-component in the system [START_REF] Heemels | Analysis of eventdriven controller for linear systems[END_REF] with the event-triggered algorithm derived from F = F iL and J = J iL , i = 1, 2, at an instant when the system is jumping, i.e. χ = x(t j ) for some t j ∈ T . In the case of linear sampled-data systems, the relations between χ, x, s, v and τ are given by

x(t k + τ ) = Γ(τ )χ, s(t k + τ ) = Kχ, v(t k + τ ) = λχ T P χ,
where Γ(τ ) = e Aτ + τ 0 e A(θ-τ ) dθBK. For the sake of simplicity, we will denote

X 1 (χ, τ ) = (Γ(τ )χ, Kχ),
for algorithm 1 and X 2 (χ, τ ) = (Γ(τ )χ, Kχ, v, τ ), for algorithm 2 for any given χ ∈ R n and T ∈ R + . Based on these linking relations, bounds on the sampling periods can be provided. This is stated in the sequel.

Proposition 4: Consider the linear system [START_REF] Heemels | Analysis of eventdriven controller for linear systems[END_REF] with the event-triggered algorithm derived from F = F iL and J = J iL , i = 1, 2. Then, the difference between two successive sampling instants is included in the interval [T i min , T i max ] defined as follows

T i max = max ρ∈R n ,|ρ|=1 max Xi(ρ,τ )∈F iL τ T i min = min ρ∈R n ,|ρ|=1 max X i (ρ,τ )∈F iL τ
Proof. Consider any state χ in R n for a solution of system [START_REF] Heemels | Analysis of eventdriven controller for linear systems[END_REF] with the event-triggered algorithm derived from F = F iL and J = J iL when a jump is occurring. By simple computations, it is clear that if, for any τ > 0, X i (χ, τ ) belongs to F iL then X i (χ/|χ|, τ ) also belongs to F iL . Then, from the definition of T i min and T i max , the next update will happen between these two bounds.

•

V. NUMERICAL EXAMPLES

A. Nonlinear example

Consider the following nonlinear system borrowed from [START_REF] Anta | To sample or not to sample: self-triggered control for nonlinear systems[END_REF], [START_REF] Byrnes | New results and examples in nonlinear feedback stabilization[END_REF]:

   ẋ1 = u 1 , ẋ2 = u 2 , ẋ3 = x 1 x 2 , ( 11
)
where (x 1 , x 2 , x 3 ) and (u 1 , u 2 ) stand respectively for the state and for the control. A stabilizing controller is computed in [START_REF] Byrnes | New results and examples in nonlinear feedback stabilization[END_REF]. It is given by, for all (x 1 , x 2 , x 3 ) in R 3 ,

u 1 (x 1 , x 2 ) = -x 1 x 2 -2x 2 x 3 -x 1 -x 3 , u 2 (x 1 , x 2 ) = 2x 1 x 2 x 3 + 3x 2 3 -x 2 . ( 12
)
A Lyapunov function for this system is computed in the same reference. It is defined by, for all (x 1 , x 2 , x 3 ) in R 3 ,

V (x) = (x 1 + x 3 ) 2 /2 + (x 2 -x 2 3 ) 2 /2 + x 2 3 .
Thus Assumption 1 holds. The simulation results are shown in Figure 2 where three different numerical simulations are done: 1) the closed-loop system in continuous-time, 2) and the event-triggered algorithms which are presented in Theorem 1, 3) and with Theorem 2. The following parameters have been selected µ(x) = 10 -3 |x| 4 , λ = 0.2. The initial conditions are x 1 (0) = -10, x 2 (0) = -5 and x 3 (0) = 5. The simulation results are shown in Figure 2. The figure contains the state, the function V , the variable τ defined in (3), representing the sampling periods and the input variable. The first and the second algorithms require respectively 20 and 14 sampling instants over a simulation time of 20sec.

B. Linear example

Consider the linear system [START_REF] Fridman | Robust sampled-data stabilization of linear systems: An input delay approach[END_REF] and the control u = Kx studied in [START_REF] Naghshtabrizi | Exponential stability of impulsive systems with application to uncertain sampled-data systems[END_REF], [START_REF] Zhang | Stability of networked control systems[END_REF] with

A = 0 1 0 -0.1 , B = 0 -0.1 , K = 3.75 11.5 T . (13)
Several robust stability conditions dedicated to the previous example of sampled-data systems can be found [START_REF] Fridman | A refined input delay approach to sampled-data control[END_REF], [START_REF] Fujioka | Stability analysis of systems with aperiodic sample-andhold devices[END_REF], [START_REF] Oishi | Stability and stabilization of aperiodic sampled-data control systems: An approach using robust linear matrix inequalities[END_REF], [START_REF] Seuret | Stability analysis for sampled-data systems with a timevarying period[END_REF]. In these articles, the main idea is to provided the largest upper-bound T so that the closed-loop system is stable for any asynchronous samplings whose period is lying in [0, T ]. It was shown in [START_REF] Seuret | Stability analysis for sampled-data systems with a timevarying period[END_REF] that the system remains stable with the upper-bound T = 1.729.

To provide an efficient event-triggered algorithm, the proposed Lyapunov matrix is taken from [START_REF] Seuret | Stability analysis for sampled-data systems with a timevarying period[END_REF] with T = 0.2 and P = 21.213 10.843 10.843 20.666 , α = 0.17.

Figure 3 shows the simulations results of the closed system using the continuous-time controller, and the event-driven Propositions 2 and 3 with ᾱ = 10 -3 and λ = 0.15. The first and the second algorithms require respectively 12 and 10 sampling instants over a simulation time of 20sec. Fig. 2. Simulation of the nonlinear system [START_REF] Hespanha | A survey of recent results in networked control systems[END_REF] with the controller [START_REF] Khalil | Nonlinear Systems[END_REF] in continuous-time (on the left) and in the sampled case following the eventtriggered algorithms given by Theorems 1 and 2 (respectively on the middle and on the right). From the top to down: time-evolution of the states, of the V -function, of the τ -variables and of the control values.

Using Proposition 4, the algorithm derived from Proposition 2 leads to the following bounds on difference between two successive sampling instants T 1 min = 0.978 and T 1 max = 6.96. Similarly, the algorithm derived from Proposition 3 leads to T 2 min = 1.02 and T 2 max = 14.91. Using both event-triggered algorithms, it is possible to use samplings whose length between two successive sampling instants could be greater than the upper-bound obtained using robust approaches from [START_REF] Fridman | A refined input delay approach to sampled-data control[END_REF], [START_REF] Fujioka | Stability analysis of systems with aperiodic sample-andhold devices[END_REF], [START_REF] Oishi | Stability and stabilization of aperiodic sampled-data control systems: An approach using robust linear matrix inequalities[END_REF], [START_REF] Seuret | Stability analysis for sampled-data systems with a timevarying period[END_REF]. This shows the main interest of the proposed method.

VI. CONCLUSIONS

In this paper using a Lyapunov-like function, two algorithms for the design of event-triggered algorithm are designed. It is assumed that a stabilizing controller for the continuous control system is given. Both event-triggered algorithms need to consider a closed-loop system with a mixed discrete/continuous dynamics (namely this is a hybrid system). Some numerical simulations illustrate the stability properties of both algorithms.

In a forecoming work, the performance issue should be analyzed. It is remarked that the event-triggered algorithms have a different performance. The first one seems to ensure a good speed of convergence on numerical simulations, whereas the second event-triggered algorithm allows less jumps and thus needs to compute less often the control variables. The advantages and disadvantages of each algorithm will be studied more precisely in a forecoming work, for a theoretical point of view (e.g. by estimating a priori the number of switches), or on applications (to understand which algorithm is better depending on the application). Fig. 3. Simulation of the linear system [START_REF] Fridman | Robust sampled-data stabilization of linear systems: An input delay approach[END_REF] and the controller u = Kx defined in (13) using the continuous-time controller (on the left) and the eventtriggered algorithms from Propositions 1 and 2 (respectively on the middle and on the right). From the top to down: time-evolution of the states, of the V -function, of the τ -variables and of the control values.

Fig. 1 .

 1 Fig. 1. Control scheme with an event-triggered algorithm.


Jian-Hua Chen

Jane L Tarry-Adkins

Kieran Matharu

Sh Giles

Susan E Yeo

Ozanne

Giles Sh Yeo

Susan E Ozanne

Maternal protein restriction

Keywords: Maternal diet, Renal function, Stress response, Senescence, Lifespan, Developmental programming Early nutrition affects renal gene expression

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Ageing is a biological process that is accompanied by a progressive decline in physiological functions leading to increased risk of diseases and ultimately death. Although many theories such as the free radical theory of ageing [START_REF] Beckman | The free radical theory of aging matures[END_REF][START_REF] Muller | Trends in oxidative aging theories[END_REF][START_REF] Vina | Theories of ageing[END_REF] provided excellent frameworks for addressing the age-long question of how and why ageing occurs, animal models have provided experimental tools to understand underlying mechanisms of the ageing process. Animal models range from worms and flies to rodents and non-human primates [START_REF] Kenyon | The plasticity of aging: insights from long-lived mutants[END_REF][START_REF] Hekimi | How genetic analysis tests theories of animal aging[END_REF][START_REF] Ladiges | Lifespan extension in genetically modified mice[END_REF][START_REF] Partridge | Mechanisms of ageing: public or private?[END_REF]. Findings from these models suggested that endocrine hormones play an important role in the regulation of ageing [START_REF] Russell | Endocrine regulation of ageing[END_REF][START_REF] Berryman | Role of the GH/IGF-1 axis in lifespan and healthspan: Lessons from animal models[END_REF]. It was first demonstrated in C. elegans that mutations in daf-2, a homologue of the mammalian insulin and insulin-like growth factor-1 (IGF1) receptors dramatically increased the lifespan of the worms [START_REF] Kenyon | A C. elegans mutant that lives twice as long as wild type[END_REF]. Subsequently extensive work showed that mutations in genes in insulin/IGF1 signaling pathways can extend the lifespan of C. elegans [START_REF] Kenyon | The plasticity of aging: insights from long-lived mutants[END_REF] and Drosophila [START_REF] Giannakou | Role of insulin-like signalling in Drosophila lifespan[END_REF]. In mammals the relationship between insulin/IGF1 signaling and lifespan is more complex due to the functional differentiation of insulin and IGF1, and the crucial role of insulin in regulation of glucose metabolism, however modest decreases in insulin or IGF1 signaling generally increase lifespan in mice [START_REF] Bartke | Minireview: role of the growth hormone/insulin-like growth factor system in mammalian aging[END_REF][START_REF] Bartke | Insulin and aging[END_REF]. It is therefore believed that these pathways of lifespan regulation are evolutionarily conserved [START_REF] Russell | Endocrine regulation of ageing[END_REF].

Ageing research in the past decade has benefited from the availability of a range of modern molecular technologies such as microarrays which allow measuring the transcriptional changes of thousands of gene simultaneously. The use of microarrays to investigate the ageing process in the above mentioned model animals has led to a better understanding of molecular mechanisms underlying not only the normal ageing process itself but also how interventions (genetic or nongenetic) can influence ageing. Comparison of gene expression profiles between young and old animals of a given species can reveal novel molecules and pathways that change with ageing and thus can help identifying biomarkers of ageing of the species under study. For example, studies in mice demonstrated that expression levels of genes involved in the inflammatory response and heat shock factors increase with age in multiple tissues and that metabolic energy pathways, extracellular matrix genes and degradation pathways generally change with age [START_REF] Lee | Gene expression profile of aging and its retardation by caloric restriction[END_REF][START_REF] Lee | Gene-expression profile of the ageing brain in mice[END_REF][START_REF] Park | Lessons learned from gene expression profile studies of aging and caloric restriction[END_REF][START_REF] Lustig | Transcriptome analysis of age-, gender-and diet-associated changes in murine thymus[END_REF]. These studies also demonstrated that caloric restriction (CR) which extends lifespan in many species attenuates many age-related changes in gene expression. In this regard, microarrays are also proven to be a useful tool in assessing the efficacy of natural or synthetic compounds in ageing interventions [START_REF] Park | Gene expression profiling of aging in multiple mouse strains: identification of aging biomarkers and impact of dietary antioxidants[END_REF]. Comparison of age-related transcriptional profiles in worms, flies, mice and humans further provided a quantitative, global view of the overall relatedness of the ageing process across different species [START_REF] Zahn | AGEMAP: a gene expression database for aging in mice[END_REF][START_REF] Kim | Common aging pathways in worms, flies, mice and humans[END_REF]. It has been demonstrated that many age-related gene expression changes are species specific, but genes in the electron transport chain pathway show common age regulation in diverse species from worms to humans [START_REF] Zahn | AGEMAP: a gene expression database for aging in mice[END_REF][START_REF] Kim | Common aging pathways in worms, flies, mice and humans[END_REF].

We have established a rodent model in which lifespan can be influenced by maternal protein restriction. We demonstrated previously that low birth weight resulting from maternal protein restriction during pregnancy followed by postnatal catch-up growth in rodents was associated with shortened lifespan, whereas protein restriction and slow growth during lactation increased lifespan [START_REF] Jennings | Early growth determines longevity in male rats and may be related to telomere shortening in the kidney[END_REF][START_REF] Ozanne | Lifespan: catch-up growth and obesity in male mice[END_REF]. We recently demonstrated that maternal restriction not only affect expression of key proteins in the muscle tissue [START_REF] Chen | Maternal protein restriction affects postnatal growth and the expression of key proteins involved in lifespan regulation in mice[END_REF] but also thymic growth in early adult life [START_REF] Chen | Early-life nutrition influences thymic growth in male mice that may be related to the regulation of longevity[END_REF] which have important implications in lifespan regulation in our model mice. Distinct from the widely known and extensively studied CR rodent models in which food is restricted at various times during postnatal life after weaning, lifespan in our rodent models is affected by maternal protein restriction during pregnancy and lactation. Unlike CR rodent models in which lifespan extension is mediated by permanent caloric restriction, we have shown that maternal protein restriction during pregnancy or lactation alone can influence lifespan. Therefore not only can the ageing process be influenced by nutritional conditions in early life but there are critical windows which can have very different impact on the ageing process later in life. The current study aims to further understand molecular mechanisms by which maternal protein restriction influences lifespan. In this study we subjected mice to exactly the same maternal protein restriction regime that is known to influence lifespan [START_REF] Ozanne | Lifespan: catch-up growth and obesity in male mice[END_REF] and examined gene expression profiles in the kidneys at weaning and albuminuria in later adult life.

Materials and Methods Animals

All animal procedures were carried out in compliance with the British Home Office Animals Act (1986). Mice (C57/b16) were bred locally at a designated animal unit of the University of Cambridge (Cambridge, UK). Adult female mice were housed individually and were maintained at 22 o C on a 12:12 hours light:dark cycle. They were mated at six weeks of age (~ 22 g in body weight) and assumed to be pregnant when a vaginal plug was expelled. Dams were fed ad libitum either a control diet (containing 20% protein) or an isocaloric low protein diet (containing 8% protein) during gestation and lactation. Both diets were purchased from Arie Blok (Woerden, The Netherlands) and detailed diet composition has been provided earlier [START_REF] Chen | Maternal protein restriction affects postnatal growth and the expression of key proteins involved in lifespan regulation in mice[END_REF]. Cross-fostering techniques were used two days after birth to establish three groups: control (offspring born to and suckled by control diet fed dams), postnatal low protein (PLP: offspring of control diet fed dams nursed by low protein diet fed dams), and recuperated (offspring of dams fed the low protein diet and nursed by control diet fed dams). A low protein diet during pregnancy had no statistically significant effect on litter size: pups per litter: low protein, 6.7 ± 0.4; control, 7.4 ± 0.4. To maximize the effects of maternal diet differences on offspring growth, recuperated pups were culled to 4 (to maximize growth during suckling) and control pups were culled to 8 (if the number of pups was greater than 8) whereas PLP pups were unculled (to reduce the plane of nutrition). Litter size standardization was carried out randomly. At 21 days of age 8 male animals per group (from 8 separate litters) were starved overnight before tissue collection. The kidneys were snap frozen in liquid nitrogen and stored at -80 o C until analysis. The remaining male offspring from all 3 groups were weaned at 21 days of age onto a standard 20% protein lab chow (LAD1, Special Diet Services, Witham, Essex, UK). Mice were housed individually and had ad libitum access to food and water. At 13 and 17 months of age 24-hour urine samples were collected. Urine collection 24-hour urine samples were collected from 13 and 17 month old mice by placing them in metabolic cages (Tecniplast, Buguggiate, Italy) for 24 hours. The volume of urine collected was recorded before the samples were frozen and stored at -80 o C until analysis.

Albumin measurements

Albumin was measured using a Mouse Albumin ELISA Quantitation Kit (Bethyl Laboratories, Texas, USA) according to the manufacturer's protocol. The plate was read at 450nm in a plate reader (Anthos Labtech HT2, Wals, Austria). Standard curves were constructed from 7.81 -500 ng/ml albumin. Extraction of total RNA and cDNA synthesis Total RNA samples were prepared using TRI reagent (Sigma, Dorset, UK), purified using RNeasy Mini Kits (Qiagen, West Sussex, UK) with a DNase digestion step included to eliminate potential contaminating DNA and quantified spectrophotometrically on a NanoDrop ND-1000 (Thermo Scientific, Waltham, MA, USA). Quality of the total RNA samples was examined by electrophoresis using an Agilent BioAnalyzer (Agilent Technologies, Queensferry, UK). First strand cDNA was reverse transcribed from 1 g of total RNA using an ImProm-II Reverse Transcription System (Promega, Hampshire, UK) with Oligo(dT) 15 as the primer according to the manufacturer's protocol.

Microarray analysis

Equal amounts of RNA extracted from each of two animals (from two litters) were combined; four combined RNA samples per group were used for microarray analysis. Sample preparation was carried out using a GeneChip® Expression 3' Amplification One-Cycle Target Labeling kit (Affymetrix, USA) according to the manufacturer's instructions. Briefly, RNA was primed with a T7 promoter oligo-dT primer and reverse transcribed to generate first-strand cDNA, which formed a template upon which second-strand cDNA was synthesized using DNA polymerase. The cDNA was then transcribed into single stranded DNA, in the presence of biotin-labeled ribonucleotide, from the T7 RNA polymerase promoter included in the double-stranded cDNA to generate antisense complementary RNA (cRNA). The biotinylated cRNA samples were then fragmented and hybridized to Affymetrix GeneChip® Mouse Genome 430 2.0 microarrays at 45 o C overnight. The hybridized array was stained with phycoerythrin-conjugated streptavidin and biotinylated antistreptavidin antibodies and scanned on an Affymetrix GeneChip Scanner 3000 7G system.

Raw image data was converted to CEL files using Affymetrix GeneChip Operating Software. All downstream analysis of microarray data was performed using GeneSpring GX 7.3 (Agilent), employing the RMA algorithm. After importing the data, each chip was normalized to the 50 th centile of the measurements taken from that chip and all gene expression data are reported as foldchange from the control state. Genes were considered up-or down-regulated at 1.5-fold. Statistical analysis was performed using a one-sample Student's t-test, looking for statistically differentially expressed genes within each condition. The test was applied to the mean of each normalized value against the baseline value of 1, where genes do not show any differential expression with respect to the control. We considered a p < 0.05 to be significant.

Gene expression analysis by real-time PCR

Real time quantitative PCR was carried out using an ABI PRISM 7300 Sequence Detection System (Applied Biosytems, Foster City, CA, USA) with a SYBR Green PCR Master Mix (Applied Biosystems, Foster City, CA, USA) and gene-specific primers. Primers were custom designed and synthesized by Sigma (Sigma, Dorset, UK) and are summarized in Table S1. 3 l of cDNA template (diluted according to relative expression level of each gene of interest) were used in a 12 l of total reaction volume in each well using a 96-well reaction plate. For every gene analyzed in this study we performed dissociation curve analysis to ensure that the primers used did not form primer-dimers. The transcripts of each gene were amplified in duplicate. Standard curves were constructed using serial diluted pooled cDNA samples. The relative expression levels were calculated against each gene's standard curve with the Ct values of each gene for each animal with the housekeeping gene cyclophilin A as a loading control. Expression of cyclophilin A showed equal levels between control, PLP and recuperated animals. Eight cDNA samples from 8 animals per group (from 8 litters) were used for real time PCR analysis.

Western blotting

Total protein was extracted from mouse kidneys by homogenization in ice-cold lysis buffer [50 mM HEPES (pH 8), 150 mM NaCl, 1% Triton X100, 1 mM sodium orthovanadate, 30 mM NaF, 10 mM sodium pyrophosphate, 10 mM EDTA, and a protease inhibitor cocktail set III (Calbiochem Novabiochem Biosciences, Nottingham, UK)]. The total protein concentration in the lysates was determined using a Sigma copper sulfate/bicinchoninic acid assay (Sigma, Dorset, UK). Samples were diluted to a concentration of 1 mg/ml in Laemmli's buffer. Twenty five micrograms of total protein were subjected to SDS-PAGE and transferred onto PVDF immobilon-P membrane (Millipore, Billerica, MA, USA). Membranes were blocked in Tris-buffered saline with 0.1% Tween-20 and 5% dry milk, and incubated with the primary antibodies against: heme oxygenase 1 (HO-1) (ab13243 Abcam, UK), Klotho (KM2076, gift from Dr Kuro-o [START_REF] Kurosu | Suppression of aging in mice by the hormone Klotho[END_REF]) and NUAK2 (ab67924 Abcam, UK; #4100 Cell Signaling Technology, Danvers, USA). The bound primary antibodies were detected by horseradish peroxidase-conjugated secondary antibodies (Dako, Glostrup, Denmark), followed by enhanced chemiluminesence (GE Healthcare, Buckinghamshire, UK). The densities of the bands were quantified using the Alpha Ease software (Alpha Innotech Corporation, San Leandro, CA, USA). Twenty four samples were run on a single gel alongside molecular weight markers and 10 and 20 g of one sample were loaded onto each gel to ensure the linearity of the signal.

Results

Body weights and kidney weights.

Protein restriction imposed during lactation significantly reduced body weights of offspring at 21 days of age whereas recuperated pups which were born small due to protein restriction during pregnancy showed significantly increased body weights (Table 1A). The difference in body weights between the three groups was maintained into later adulthood although pups of all three groups were weaned onto a standard 20% protein lab chow and had ad libitum access to food and water. Thus at 13 and 17 months of age body weights of PLP mice were significantly lighter than controls (p < 0.001) whereas recuperated mice were heavier (p = 0.12 for 13 months old and p < 0.05 for 17 months old) (Table 1B). Kidneys of PLP offspring at 21 days of age were significantly lighter (p < 0.001) than controls, however kidneys of recuperated offspring were not statistically different to controls (Table 1A). In terms of relative organ weights both PLP and recuperated kidneys were significantly lower than controls (Table 1A). Urinary albumin at 13 and 17 months of age At 13 months of age there was no difference between control and recuperated mice but PLP mice had significantly reduced albuminuria than the control (p < 0.05) and recuperated offspring (p < 0.001) (Figure 1). There was a tendency for recuperated mice to have increased albuminuria than controls at 13 months of age (p = 0.06). At 17 months of age albumin levels of control and recuperated mice were higher than those of 13 month-old mice with the increase in recuperated mice being statistically significant (p < 0.05), however this age-related increase was not observed in PLP mice (Figure 1). Thus PLP mice at 17 months of age had significantly reduced albuminuria compared to control (p < 0.01) and recuperated offspring (p < 0.001) (Figure 1).

Gene expression profiles in the kidneys

Affymetrix GeneChip Mouse Genome 430 V2.0 Array analysis revealed that 36 genes in PLP and 52 genes in recuperated kidneys were up-regulated compared to controls, whereas 54 genes in PLP and 302 genes in recuperated kidneys were down-regulated compared to control tissues (Figure 2). Tables 2 and3 show, respectively, the top 25 genes that were up-and down-regulated in expression in the PLP and recuperated kidneys compared to the controls. Maternal protein restriction imposed during lactation (PLP) influenced gene expression very differently from maternal protein restriction that was imposed only during pregnancy (recuperated) both for up-and down-regulated genes. Thus there was very little overlap for genes in PLP or recuperated kidneys that were either up-or downregulated by maternal protein restriction (Tables 2 and3; for complete gene lists see supplementary Excel files). Heme oxygenase 1 (HO-1) was the most up-regulated and Cyp27b1 was the most down-regulated gene in the PLP kidneys (Tables 2 and3). NUAK2 was the most up-regulated gene in the recuperated kidneys (Table 2). Among the down-regulated genes in the recuperated kidneys that have implications in the regulation of the ageing process were Lon peptidase, FOXO3 (Forkhead box O3), SOD1 (copper-zinc superoxide dismutase) and Sestrins (Table 3 and Supplementary Excel files).

Pathway analysis

We further analyzed the data using a pathway analysis package (Ingenuity Pathway Analysis) that detects groups of functionally related annotated genes. Table 4 shows the top 5 pathways with the genes that were up-or down-regulated in the PLP and recuperated kidneys compared to controls. This analysis also showed no overlapping pathways between the PLP and recuperated kidneys. Gene function category analysis showed that genes whose expression was influenced by maternal diet are involved in diverse functions including cell growth and metabolism (Tables S1 andS2).

PCR validation

Real time quantitative PCR was performed to validate the microarray data using 8 cDNA samples prepared from 8 animals (from 8 litters) in each group (control, PLP and recuperated). Primer S3. PCR analysis showed that fold changes for genes selected from the microarray gene lists were similar to those detected by microarray analysis for up-regulation in both PLP and recuperated kidneys and down-regulation in PLP kidneys (Tables 2 and3). Some of the down-regulated genes in recuperated kidneys did not validate, probably due to already low levels of expression, which is similar to what others observed with PCR validation for down-regulated genes [START_REF] Lustig | Transcriptome analysis of murine thymocytes reveals age-associated changes in thymic gene expression[END_REF]. In general, PCR analysis suggests that the gene expression data generated by microarray analysis were robust and reproducible.

The significant down-regulation of cytochrome P450 27b (Cyp 27b1) gene expression in the PLP kidneys prompted us to investigate Klotho gene expression as Cyp 27b1 can be regulated by Klotho [START_REF] Kuro-O | Klotho and aging[END_REF], which is an important anti-ageing hormone [START_REF] Kurosu | Suppression of aging in mice by the hormone Klotho[END_REF]. However, real time PCR analysis showed no difference in Klotho gene expression between the three groups (data not shown).

Protein expression of HO-1, NUAK2 and Klotho.

HO-1 protein expression was significantly up-regulated in PLP kidneys whereas no difference was observed in recuperated kidneys (Figure 3A). NUAK2 protein expression could not be detected by Western blotting using two different commercially available antibodies. The significant downregulation of cytochrome P450 27b (Cyp 27b1) gene expression in the PLP kidneys also prompted us to investigate Klotho protein expressions in the kidneys. Klotho protein expression showed no significant difference between the three groups (Figure 3B).

Discussion

In this study we investigated albuminuria in old age (13 and 17 months old) and gene expression profiles in the kidneys at a young age (21 days old) of our maternal protein restriction mouse models with an aim to understand how maternal protein restriction influence lifespan. The kidneys are particularly affected by age [START_REF] Martin | Renal ageing[END_REF] and age-associated nephropathy is known to be one of the important non-neoplastic lesions that could affect overall survival and longevity [START_REF] Razzaque | Does renal ageing affect survival?[END_REF]. Urinary albumin excretion has been recognized as the "gold standard" of renal impairment and serves as an important diagnostic tool as a marker of glomerular dysfunction [START_REF] Hong | Markers of diabetic nephropathy[END_REF] and is widely accepted as a risk factor for the development of renal disease [START_REF] Birn | Renal albumin absorption in physiology and pathology[END_REF]. Albumin is a large molecular weight plasma protein which under normal conditions is filtered in small amounts through the glomerulus and later re-absorbed by the renal tubular cells and metabolized. However, damage to the renal glomerular basement membrane can allow albumin to pass through the glomerulus and be excreted in the urine. Our albuminuria data suggests that the recuperated mice may have elevated levels of glomerular damage compared to control animals with age, whereas the PLP offspring are protected from glomerular injury. As the actual nepheron number is not assessed in this study it is difficult to know whether a reduction in nephron number resulting from protein restriction during pregnancy [START_REF] Hoppe | Effects of dietary protein restriction on nephron number in the mouse[END_REF], compounded with potential compensatory hypertrophy [START_REF] Moritz | Kidney development and the fetal programming of adult disease[END_REF] that may occur due to catch-up growth during lactation contribute to the age-dependent increase in albuminuria in the recuperated animals in late adult life. However whatever the causes of the observed differences in albuminuria data, they are consistent with the known differences in lifespan.

To understand how maternal protein restriction impacts on long-term glomerular dysfunction which might in turn influence lifespan, we chose to investigate global gene expression profiles in the kidneys of 21 day old offspring. This would allow us to reveal genes and/or pathways that are influenced by maternal diet which may mediate programming effects of early life events on later life health and longevity. Microarray analysis showed that gene expression in the kidney was broadly affected by maternal diet and that maternal protein restriction during lactation affected expression of genes that were different to those affected by maternal protein restriction during pregnancy. Interestingly the numbers of genes that were up-and down-regulated in recuperated kidneys were greater than those in PLP kidneys. This suggests that in utero growth restriction coupled with catch-up growth during early postnatal life had a greater impact on the kidney than that of slow postnatal growth caused by maternal protein restriction during lactation. This is in line with the previous longevity data which showed that the extent of decrease in lifespan in recuperated mice was greater than that of increased lifespan in PLP mice [START_REF] Ozanne | Lifespan: catch-up growth and obesity in male mice[END_REF].

The fold changes in expression of the genes affected by maternal diet were relatively modest (< 3.6 fold). In addition, the number of affected genes that fall into a particular pathway were small. This suggests that maternal diet imposes programming effects on ageing through small changes in expression of many genes that can influence multiple pathways that will eventually have significant cumulative effects on tissue function and thus ultimately on lifespan. This is in line with the notion that functional decline during the ageing process is not the result of the complete failure of a small number of cellular processes but rather, it is the slight weakening of many pathways that cumulatively causes a significant decrease in cell function [START_REF] Rodwell | A transcriptional profile of aging in the human kidney[END_REF]. This notion is based on the observation that transcriptional differences between young and old individuals involve an accumulation of small changes in expression from many genes rather than large expression changes in a small number of genes [START_REF] Rodwell | A transcriptional profile of aging in the human kidney[END_REF].

The up-regulation of heme oxygenase 1 (HO-1) in the kidneys of long-lived PLP mice suggests that a cytoprotective pathway involving this enzyme is enhanced. HO-1 is an enzyme that catalyzes the rate-limiting step in heme degradation resulting in the formation of iron, carbon monoxide and biliverdin [START_REF] Galbraith | Heme oxygenase: who needs it?[END_REF]. Biliverdin can then be rapidly reduced to bilirubin by biliverdin reductase [START_REF] Galbraith | Heme oxygenase: who needs it?[END_REF]. Both biliverdin and bilirubin have antioxidant effects while carbon monoxide and iron have anti-inflammatory and antiapoptotic effects respectively [START_REF] Courtney | Heme oxygenase 1: does it have a role in renal cytoprotection?[END_REF]. Indeed, many studies have shown that up-regulation of HO-1 can bring about beneficial effects. For example, up-regulation of HO-1 in islet cells results in protection from apoptosis and improved in vivo function after A c c e p t e d M a n u s c r i p t Licenced copy. Copying is not permitted, except with prior permission and as allowed by law.

© 2010 The Authors Journal compilation © 2010 Portland Press Limited transplantation [START_REF] Pileggi | Heme oxygenase-1 induction in islet cells results in protection from apoptosis and improved in vivo function after transplantation[END_REF]. Induction of HO-1 improves aortic endothelia dysfunction in insulin resistant rats [START_REF] Chen | Hemin, a heme oxygenase-1 inducer, improves aortic endothelial dysfunction in insulin resistant rats[END_REF] and protects against intestinal ischemia/reperfusion injury [START_REF] Wasserberg | Heme oxygenase-1 upregulation protects against intestinal ischemia/reperfusion injury: a laboratory based study[END_REF]. The cytoprotective role of HO-1 in renal function regulation including prevention of renal injury and vascular disease is particularly well documented [START_REF] Abraham | Heme oxygenase: the key to renal function regulation[END_REF]. The fact that induction of HO-1 occurs as a beneficial response to various stresses [START_REF] Jarmi | Heme oxygenase and renal disease[END_REF] may suggest that the postnatal low protein feeding regime can induce a beneficial stress response which promotes tissue function and longevity. This is in agreement with the hormesis hypothesis which proposes that mild stress-induced stimulation of protective mechanisms in cells and organisms results in long-term biologically beneficial effects [START_REF] Rattan | Hormesis in aging[END_REF]. This may explain why albuminuria is reduced in old PLP mice. Indeed our previous studies in the rat showed protein restriction during lactation confers nephroprotective effects which is associated with increased expression of antioxidant enzymes and decreased age-dependent renal telomere shortening [START_REF] Tarry-Adkins | Protein restriction in lactation confers nephroprotective effects in the male rat and is associated with increased anti-oxidant expression[END_REF].

NUAK2 is a member of the AMP-activated protein kinase catalytic subunit family [START_REF] Lefebvre | Identification and characterization of a novel sucrose-non-fermenting protein kinase/AMP-activated protein kinase-related protein kinase, SNARK[END_REF]. NUAK2 can be induced by various cellular stresses such as ER stress, elevation of cellular AMP, hyper-osmotic stress, and ultraviolet light [START_REF] Lefebvre | Regulation of SNARK activity in response to cellular stresses[END_REF]. A recent study showed that up-regulation of NUAK2 can induce premature cellular senescence [START_REF] Humbert | Regulation of ploidy and senescence by the AMPK-related kinase NUAK1[END_REF]. Increased expression of NUAK2 in the kidneys of recuperated mice may therefore suggest that in utero growth restriction followed by catch-up growth can result in accelerated cellular senescence. Indeed we previously observed that poor maternal nutrition followed by accelerated postnatal growth in the rat leads to accelerated telomere shortening and increased markers of cell senescence in rat islets [START_REF] Tarry-Adkins | Poor maternal nutrition followed by accelerated postnatal growth leads to telomere shortening and increased markers of cell senescence in rat islets[END_REF].

Among the genes that were down-regulated are genes that also have implications in regulation of the ageing process. The most down-regulated gene in PLP kidneys was Cyp27b1. The Cyp27b1 gene encodes 1 -hydroxylase, the key enzyme that synthesizes active vitamin D (1, 25dihydroxyvitamin D) in the kidney. It is known that Klotho which is predominantly expressed in the kidney promotes longevity through regulation of vitamin D metabolism and phosphate homeostasis [START_REF] Kuro-O | Klotho and aging[END_REF]. Expression of Cyp27b1 in kidney is suppressed by Klotho in conjunction with FGF23 [START_REF] Kuro-O | Klotho and aging[END_REF]. High concentrations of active vitamin D are observed in Klotho mutant mice that undergo premature ageing because of the increased expression of the renal 1 -hydroxylase gene [START_REF] Yoshida | Mediation of unusually high concentrations of 1,25-dihydroxyvitamin D in homozygous klotho mutant mice by increased expression of renal 1alpha-hydroxylase gene[END_REF]. In contrast, many deleterious phenotypic features such as abnormal mineral ion metabolism and softtissue calcification of Klotho-deficient mice can be reversed by deletion of vitamin D 1hydroxylase [START_REF] Ohnishi | Reversal of mineral ion homeostasis and soft-tissue calcification of klotho knockout mice by deletion of vitamin D 1alpha-hydroxylase[END_REF]. Although we did not detect an increase in Klotho expression at the mRNA or protein levels the down-regulation in Cyp27b1 in PLP kidneys suggest that vitamin D metabolism and phosphate homeostasis may shift towards those in the Klotho transgenic mice that live longer [START_REF] Kurosu | Suppression of aging in mice by the hormone Klotho[END_REF]. In contrast, genes that are implicated in promoting longevity such as Lon peptidase, FOXO3, SOD1 and Sestrin1 were down-regulated in the kidneys of recuperated mice. Lon peptidase plays an important role in eliminating oxidatively modified intracellular proteins thus contributing to protein quality control and homeostasis [START_REF] Petropoulos | Maintenance of proteins and aging: the role of oxidized protein repair[END_REF]. Lon protease expression and activity declines with age and this is thought to contribute to the accumulation of the oxidized protein aggregates typically observed in ageing and diseased cells [START_REF] Ngo | Importance of the lon protease in mitochondrial maintenance and the significance of declining lon in aging[END_REF]. FOXO3 is a member of a transcription factor family known to extend lifespan in invertebrates [START_REF] Daitoku | FOXO transcription factors in the regulatory networks of longevity[END_REF]. FOXO transcription factors are regulated by growth factors and hormones and control diverse cellular functions, including proliferation, differentiation, apoptosis, DNA repair, defense against oxidative stress and ageing [START_REF] Sedding | FoxO transcription factors in oxidative stress response and ageing--a new fork on the way to longevity?[END_REF]. SOD1 is one of the target genes of FOXO transcription factors and functions as a vital enzyme in metabolizing ROS. Thus down-regulation of FOXO3 and SOD1 suggests that the antioxidant defense system in recuperated mice is impaired. Sestrin1 is a member of the sestrin family which exhibit oxidoreductase activity and may function as antioxidants [START_REF] Budanov | Regeneration of peroxiredoxins by p53-regulated sestrins, homologs of bacterial AhpD[END_REF]. It has been shown that repression of sestrin family genes can lead to ROS up-regulation and genetic instability [START_REF] Kopnin | Repression of sestrin family genes contributes to oncogenic Ras-induced reactive oxygen species up-regulation and genetic instability[END_REF]. A recent study showed that sestrin functions as a feedback inhibitor of TOR (target of rapamycin) and prevents age-related pathologies resulting from diminished autophagic clearance of damaged mitochondria, protein aggregates, or lipids [START_REF] Lee | Sestrin as a feedback inhibitor of TOR that prevents age-related pathologies[END_REF].

In summary, the present study demonstrates that maternal diet can have long-lasting effects on albuminuria in our maternal protein restricted mouse model; albuminuria in long-lived PLP mice was ameliorated whereas in short-lived recuperated mice albuminuria was accelerated. Microarray

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  Equal loading was confirmed by Coomassie blue staining. Statistical analysis Clinical Science Immediate Publication. Published on 08 Jun 2010 as manuscript CS20100230 THIS IS NOT THE VERSION OF RECORD -see doi:10.1042/CS20100230 A c c e p t e d M a n u s c r i p t Licenced copy. Copying is not permitted, except with prior permission and as allowed by law. © 2010 The Authors Journal compilation © 2010 Portland Press Limited Data were analyzed by ANOVA with maternal diet and age (where appropriate) as the independent 210 variables. Duncan's post hoc testing was used to analyze significant differences between groups 211 where appropriate. A p < 0.05 was considered statistically significant. For urinary albumin level 212 analysis non-parametric Kruskal-Wallis tests followed by Dunn's post tests were used for group 213 comparisons, and Mann-Whitney tests for age comparisons. 214 Clinical Science Immediate Publication. Published on 08 Jun 2010 as manuscript CS20100230 THIS IS NOT THE VERSION OF RECORD -see doi:10.1042/CS20100230 A c c e p t e d M a n u s c r i p t Licenced copy. Copying is not permitted, except with prior permission and as allowed by law. © 2010 The Authors Journal compilation © 2010 Portland Press Limited

Clinical

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Clinical

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Clinical

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Clinical

  Science Immediate Publication. Published on 08 Jun 2010 as manuscript CS20100230 THIS IS NOT THE VERSION OF RECORD -see doi:10.1042/CS20100230 A c c e p t e d M a n u s c r i p t Licenced copy. Copying is not permitted, except with prior permission and as allowed by law. © 2010 The Authors Journal compilation © 2010 Portland Press Limited analysis revealed that expression of a wide range of genes in the kidneys at 21 days of age was 391 affected by maternal protein restriction. It suggested that tissues of the long-lived mice are equipped 392 with a better cytoprotective function which ultimately promotes longevity. In contrast, the 393 antioxidant defense system and protein quality control/homeostasis in tissues of short-lived 394 recuperated mice may be compromised which could lead to increased oxidative stress, resulting in 395 accelerated cellular senescence and ultimately reduced lifespan. The detection of these changes in 396 genes implicated in the regulation of ageing provides insight into molecular mechanisms by which 397 maternal diet and early growth rate influences the ageing trajectory. 398 Clinical Science Immediate Publication. Published on 08 Jun 2010 as manuscript CS20100230 THIS IS NOT THE VERSION OF RECORD -see doi:10.1042/CS20100230 A c c e p t e d M a n u s c r i p t Licenced copy. Copying is not permitted, except with prior permission and as allowed by law. © 2010 The Authors Journal compilation © 2010 Portland Press Limited

Figure Legends Figure 1 .

 1 Figure LegendsFigure 1. Median albumin in urine samples collected from 13 and 17 months old mice. Error bars are interquartile ranges. Kruskal-Wallis: p < 0.001 at 13 and 17 months old. Dunn's post test: * p < 0.05, ** p < 0.01 compared to control. ‡ ‡ ‡ p < 0.001 compared to recuperated. Mann-Whitney test: + p < 0.05 compared to 13 months.Figure 2. Number of up-and down-regulated genes in the kidneys of postnatal low protein (PLP) and recuperated animals compared to controls. Light shaded bars depict the number of genes which increased expression levels and dark shaded bars depict the number of genes which decreased expression levels.Figure 3. Protein expression of HO-1 and Klotho in 21 day old kidneys of control, PLP and recuperated mice. (A) 25 g of total protein were analyzed by Western blotting using antibodies as indicated. Equal loading was confirmed by Coomassie blue staining. (B) Densitometry analysis data for HO-1 protein expression and (C) Densitometry analysis data for Klotho protein expression were expressed as arbitrary unit (AU).

Figure 2 .

 2 Figure LegendsFigure 1. Median albumin in urine samples collected from 13 and 17 months old mice. Error bars are interquartile ranges. Kruskal-Wallis: p < 0.001 at 13 and 17 months old. Dunn's post test: * p < 0.05, ** p < 0.01 compared to control. ‡ ‡ ‡ p < 0.001 compared to recuperated. Mann-Whitney test: + p < 0.05 compared to 13 months.Figure 2. Number of up-and down-regulated genes in the kidneys of postnatal low protein (PLP) and recuperated animals compared to controls. Light shaded bars depict the number of genes which increased expression levels and dark shaded bars depict the number of genes which decreased expression levels.Figure 3. Protein expression of HO-1 and Klotho in 21 day old kidneys of control, PLP and recuperated mice. (A) 25 g of total protein were analyzed by Western blotting using antibodies as indicated. Equal loading was confirmed by Coomassie blue staining. (B) Densitometry analysis data for HO-1 protein expression and (C) Densitometry analysis data for Klotho protein expression were expressed as arbitrary unit (AU).

Figure 3 .

 3 Figure LegendsFigure 1. Median albumin in urine samples collected from 13 and 17 months old mice. Error bars are interquartile ranges. Kruskal-Wallis: p < 0.001 at 13 and 17 months old. Dunn's post test: * p < 0.05, ** p < 0.01 compared to control. ‡ ‡ ‡ p < 0.001 compared to recuperated. Mann-Whitney test: + p < 0.05 compared to 13 months.Figure 2. Number of up-and down-regulated genes in the kidneys of postnatal low protein (PLP) and recuperated animals compared to controls. Light shaded bars depict the number of genes which increased expression levels and dark shaded bars depict the number of genes which decreased expression levels.Figure 3. Protein expression of HO-1 and Klotho in 21 day old kidneys of control, PLP and recuperated mice. (A) 25 g of total protein were analyzed by Western blotting using antibodies as indicated. Equal loading was confirmed by Coomassie blue staining. (B) Densitometry analysis data for HO-1 protein expression and (C) Densitometry analysis data for Klotho protein expression were expressed as arbitrary unit (AU).

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Acknowledgments

We thank A Wayman, A Flack and D Hawkes for their technical expertise. We are grateful to Dr M Kuro-o for the generous gift of the anti-Klotho antibody. Funding This work was supported by the Biotechnology and Biological Sciences Research Council [grant number BB/E002161/1], the British Heart Foundation [grant number FS/09/029/27902], the Medical Research Council [grant number G0600717]. SEO is a member of the Medical Research Council Centre for Obesity and Related Metabolic Diseases (MRC CORD), and of the University of Cambridge Metabolic Research Laboratories.