Laura Dal Col
email: ldalcol@laas.fr

Sophie Tarbouriech

Luca Zaccarian
email: zaccarian@laas.fr

Michel Kieffer
email: kieffer@lss.supelec.fr.

Equivalent Conditions for Synchronization of Identical Linear Systems and Application to Quality-Fair Video Delivery

come

I. INTRODUCTION

In recent years, consensus and synchronization problems among multi-agent systems have received an intensive interest in the literature, due to the variety of applications in many different areas including cooperative control of unmanned areal vehicles, formation control of mobile robots and communication among sensor networks. See, e.g., [START_REF] Fax | Information flow and cooperative control of vehicle formations[END_REF], [START_REF] Hara | Stability analysis of linear systems with generalized frequency variables and its applications to formation control[END_REF], [START_REF] Jadbabaie | Coordination of groups of mobile autonomous agents using nearest neighbor rules[END_REF], [START_REF] Kim | Stabilization of multi-agent dynamical systems for cyclic pursuit behavior[END_REF], [START_REF] Olfati-Saber | Consensus and cooperation in networked multi-agent systems[END_REF], [START_REF] Ren | Information consensus in multivehicle cooperative control[END_REF]. Specifically, consensus refers to agents coming to a global agreement on a state value, by the exchange of information modeled by some communication graph. It has been shown in [START_REF] Moreau | Stability of continuous-time distributed consensus algorithms[END_REF], [START_REF] Moreau | Stability of multiagent systems with time-dependent communication links[END_REF] that mild assumptions on graph connectivity ensure to uniformly exponentially reach consensus, see also [START_REF] Jadbabaie | Coordination of groups of mobile autonomous agents using nearest neighbor rules[END_REF], [START_REF] Olfati-Saber | Consensus problems in networks of agents with switching topology and time-delays[END_REF]. Compared to consensus problems, the synchronization literature refers to agents moving toward a common trajectory in the configuration space [START_REF] Hale | Diffusive coupling, dissipation, and synchronization[END_REF], [START_REF] Sepulchre | Consensus on nonlinear spaces[END_REF], [START_REF] Slotine | A study of synchronization and group cooperation using partial contraction theory[END_REF], [START_REF] Stan | Analysis of interconnected oscillators by dissipativity theory[END_REF].

Consensus algorithms are primarily studied when the agents' open-loop internal dynamics are described by an integrator chain (e.g. single-or double-integrator models [START_REF] Olfati-Saber | Consensus problems in networks of agents with switching topology and time-delays[END_REF], [START_REF] Ren | Consensus algorithms for double-integrator dynamics[END_REF]). Recently, for full generality, the consensus problem has been investigated considering agents modeled by general linear time-invariant (LTI) systems [START_REF] Seo | Consensus of high-order linear systems using dynamic output feedback compensator: low gain approach[END_REF], [START_REF] Wieland | On consensus in multi-agent systems with linear high-order agents[END_REF], [START_REF] Yang | Constructing consensus controllers for networks with identical general linear agents[END_REF]. Consensus and synchronization problems are extensively studied in the literature for identical multi-agent systems, see, e.g., [START_REF] Fax | Graph Laplacians and stabilization of vehicle formations[END_REF], [START_REF] Fax | Information flow and cooperative control of vehicle formations[END_REF], [START_REF] Ren | Consensus strategies for cooperative control of vehicle formations[END_REF], [START_REF] Wang | Consensus of multi-agent linear dynamic systems[END_REF]. In [START_REF] Wang | Finite-time consensus problems for networks of dynamic agents[END_REF] a state-feedback consensus protocol is proposed for linear multi-agent systems with switching topology.

In [START_REF] Xiao | Consensus problems for high-dimensional multiagent systems[END_REF] sufficient and necessary conditions on consensus of linear multi-agent systems are provided by using the full state information of the agents. In [START_REF] Tuna | LQR-based coupling gain for synchronization of linear systems[END_REF], a linear quadratic regulator (LQR) based optimal control approach was used for the controller design via state-feedback information. When the full state is not available, an observer can be used to estimate the states [START_REF] Hong | Distributed observers design for leader-following control of multi-agent networks[END_REF], [START_REF] Li | Leader-follower consensus of multiagent systems[END_REF], [START_REF] Scardovi | Synchronization in networks of identical linear systems[END_REF], which makes the control architecture more complex. To overcome this problem, output-feedback based control may provide satisfactory solutions and some methods have been given [START_REF] Li | Consensus of multiagent systems and synchronization of complex networks: a unified viewpoint[END_REF], [START_REF] Liu | Dynamic output feedback control for consensus of multi-agent systems: An h-∞; approach[END_REF], [START_REF] Zhang | Optimal design for synchronization of cooperative systems: State feedback, observer and output feedback[END_REF]. A low gain approach to dynamic output feedback compensator design for consensus was given in [START_REF] Seo | Consensus of high-order linear systems using dynamic output feedback compensator: low gain approach[END_REF]. Reference [START_REF] Li | Consensus of multiagent systems and synchronization of complex networks: a unified viewpoint[END_REF] proposed an observer-type consensus protocol designed using Finsler's lemma and LMI techniques. Consensus analysis and design via LMI numerical procedures are presented in [START_REF] Seuret | Consensus under communication delays[END_REF], [START_REF] Sun | Average consensus in directed networks of dynamic agents with time-varying communication delays[END_REF], [START_REF] Xi | Consensus analysis and design for highorder linear swarm systems with time-varying delays[END_REF] for interconnected systems under communication delays.

Despite the above mentioned extensive amount of work in the field, one cannot find a general theorem about synchronization of identical linear systems without any structure. Since for synchronization solutions may diverge while synchronizing, one should pay special attention to the fact that the attractor (the synchronization set where all the states concide) is unbounded, and then uniformity of stability and attractivity is nontrivially shown. Establishing equivalent conditions for this property, also involving strict Lyapunov functions, is the first contribution of this paper.

A second contribution of this paper is the use of the necessary and sufficient synchronization conditions mentioned above to the design of a quality-fair video streaming system to several users sharing some common wireless ressource, considering the model proposed in [START_REF] Changuel | Control of multiple remote servers for quality-fair delivery of multimedia contents[END_REF], [START_REF] Changuel | Control of distributed servers for quality-fair delivery of multiple video streams[END_REF]. When several users share some communication link to get streamed video contents, simple bit-rate or bandwidth fair allocation strategies are usually inappropriate. Such strategies are agnostic of the rate-quality characteristics of the delivered contents. Rather static video contents such as news may be efficiently delivered with a moderate bit rate, that would be insufficient to enjoy an action motion picture of decent quality. This has motivated the recent development of quality-fair video delivery techniques, such as [START_REF] Changuel | Joint encoder and buffer control for statistical multiplexing of multimedia contents[END_REF], [START_REF] Cho | Utility max-min flow control using sloperestricted utility functions[END_REF], [START_REF] Cicalo | Cross-layer algorithms for distortion-fair scalable video delivery over OFDMA wireless systems[END_REF], [START_REF] Li | Content-aware distortionfair video streaming in congested networks[END_REF]. For example, [START_REF] Cho | Utility max-min flow control using sloperestricted utility functions[END_REF] considers an utility max-min fair resource allocation, which tries to maximize the worst utility. Nevertheless, it does not consider the temporal variability of the rate-utility characteristics of the contents, or the delays introduced by the network and the buffers of the delivery system. In [START_REF] Li | Content-aware distortionfair video streaming in congested networks[END_REF], a content-aware distortion-fair video delivery scheme is proposed, assuming that the characteristics of the video frames are known in advance, which restricts its usage to the streaming of stored videos. In [START_REF] Cicalo | Cross-layer algorithms for distortion-fair scalable video delivery over OFDMA wireless systems[END_REF], a Lagrangian optimization framework is considered to maximize the sum of the achievable rates while minimizing the distortion difference among streams. This requires to gather all rate-utility characteristics of the streams at the control unit. The user experience is accurately modeled in [START_REF] Chen | Rate adaptation and admission control for video transmission with subjective quality constraints[END_REF] using the empirical cumulative distribution function of the predicted video quality. Feedback control techniques have been considered in [START_REF] Changuel | Control of multiple remote servers for quality-fair delivery of multimedia contents[END_REF], [START_REF] Changuel | Control of distributed servers for quality-fair delivery of multiple video streams[END_REF] to reach a quality fairness among users, while controlling the level of the buffers in the network or the buffering delay. However, the tuning of the parameters is nontrivial and has been performed heuristically in these works.

To the best of our knowledge, it is the first time that such parallel video delivery problem with ressource and qualityfaireness constraints is studied with consensus techniques. Our proposed approach leads to an attractive linear static output feedback design problem that ensures suitable performance guarantees in addition to consensus. Preliminary results in this direction have been reported in [START_REF] Col | A linear consensus approach to quality-fair video delivery[END_REF]. We include here a more detailed description of our approach, proofs that were missing in [START_REF] Col | A linear consensus approach to quality-fair video delivery[END_REF], a systematic synthesis procedure for the linear static output feedback design problems, and revised simulation tests.

Summarizing, the contribution of this paper, is twofold. First, we derive a general Lyapunov-based synchronization result for continuous-and discrete-time identical linear systems. More precisely, the equivalence is proven among several well known conditions and uniform global exponential stability of the consensus set. Second, these results are shown to lead to a systematic design technique based on iterative LMIs for an optimized selection of the PI gains in the experimental application modeled in [START_REF] Changuel | Control of multiple remote servers for quality-fair delivery of multimedia contents[END_REF], [START_REF] Changuel | Control of distributed servers for quality-fair delivery of multiple video streams[END_REF] maximizing the convergence rate to the synchronization set while guaranteeing it uniform global exponential stability. A further illustration of our results is given in our simulation section, where we compare our results to the ones obtained with the PI tuning of [START_REF] Changuel | Control of multiple remote servers for quality-fair delivery of multimedia contents[END_REF], [START_REF] Changuel | Control of distributed servers for quality-fair delivery of multiple video streams[END_REF] which followed a heuristic technique. Our results on necessary (and sufficient) stability conditions predict that the heuristic selection of [START_REF] Changuel | Control of distributed servers for quality-fair delivery of multiple video streams[END_REF] is non stabilizing, and indeed simulation results show evidence of instability that disappears with our selection.

The paper is organized as follows. Section II presents the linear consensus problem and gives the main result. In Section III the quality-fair video delivery problem is cast as a distributed consensus problem. In Section IV a systematic method to design the controller gains is proposed using Finsler's lemma and LMI techniques. The effectiveness of this method is illustrated on experimental tests in Section V. Concluding remarks end the paper.

Notation. We use x + = x + (j) = x(j + 1) to denote the push-forward operator, ∀j ∈ Z + , x d = x d (j) = x(j -1) to denote the one step delay operator, and x dd = x dd (j) = x(j -2) to denote the two steps delay operator. We denote with 1 N the N dimensional (column) vector, for which entries are all 1. For any square matrices A 1 , . . . , A N , the notation diag (A 1 , . . . , A N ) indicates the block diagonal matrix whose diagonal blocks are A 1 , . . . , A N . We denote with C ≤β the set of the complex numbers with modulus less or equal to β. The symbol 0 m,n denotes the zero matrix of size m × n

II. NECESSARY AND SUFFICIENT CONDITIONS FOR SYNCHRONIZATION OF IDENTICAL LINEAR SYSTEMS

Consider N identical dynamical systems, governed by:

δx i = Ax i + Bu i y i = Cx i i = 1, . . . , N (1)
where δx = ẋ for continuous-time and δx = x + for discretetime. In (1),

x i ∈ R n , u i ∈ R, y i ∈ R. Consider the interconnection: u = -Ly, (2)
where

u = [u 1 . . . u N ] ∈ R N , y = [y 1 . . . y N ] ∈ R N and L = L ∈ R N ×N is the graph Laplacian of the network. Also denote the eigenvalues of L as 0 = λ 0 ≤ λ 1 ≤ • • • ≤ λ N -1
, where it is emphasized (see [START_REF] Fax | Information flow and cooperative control of vehicle formations[END_REF]) that L has always an eigenvalue at zero, that corresponds to the eigenvector 1 N .

With the goal of establishing synchronization among systems (1), let us introduce the following consensus set:

A := {x : x i -x j = 0, ∀i, j ∈ {1, . . . , N }} .

Let also recall that given a set X , |x| X = inf y∈X |x -y|.

Our main consensus result, given below, relies on suitable properties of quadratic functions with respect to set A in (3). These are given in the next lemma, whose proof is reported in the appendix. Lemma 1: Consider any unitary matrix T ∈ R N ×N whose first column is given by 1 √ N 1 N and the diagonal matrix ∆ = I N -e 1 e 1 , where e 1 = [1 0 . . . 0] ∈ R N is the first element of the Euclidean basis. Then there exist scalars c 1 , c1 , c 2 , c2 > 0 such that for any n ∈ N and any x ∈ R N n , where x k ∈ R n , ∀k = 1, . . . , N , we have:

c1 |x| 2 A = c 1 N k=2 |x 1 -x k | 2 ≤ x (T ∆T ⊗ I n )x, (4a) c2 |x| 2 A = c 2 N k=2 |x 1 -x k | 2 ≥ x (T ∆T ⊗ I n )x. (4b)
Based on Lemma 1, we can now state a set of necessary and sufficient conditions for synchronization of identical linear systems. The proof combines the stability results in [START_REF] Fax | Information flow and cooperative control of vehicle formations[END_REF] with the established output feedback coupling approach of [START_REF] Scardovi | Synchronization in networks of identical linear systems[END_REF].

Parts of the following result can be found in the literature: for example, implication (i) =⇒ (iii) is established in an equivalent formulation in [START_REF] Fax | Information flow and cooperative control of vehicle formations[END_REF]Theorem 3], [START_REF] Xia | Synchronization conditions for diffusively coupled linear systems[END_REF]Theorem 1] and [START_REF] Seo | Consensus of high-order linear systems using dynamic output feedback compensator: low gain approach[END_REF]Theorem 1] for the convergence part. In [START_REF] Wieland | An internal model principle is necessary and sufficient for linear output synchronization[END_REF], the equivalence (i) ⇐⇒ (iv) is proven for the continuous-time case.

Theorem 1: Consider the continuous-time [respectively, discrete-time] system (1), (2) and the attractor A in (3). The following statements are equivalent:

(i) Matrices

A k := A -λ k BC, k = 1, . . . , N -1 (5)
are Hurwitz [respectively, Schur-Cohn]. (ii) There exists a strict quadratic Lyapunov function V (x) satisfying:

c1 |x| 2 A ≤ V (x) ≤ c2 |x| 2 A , (6a)
V (x) ≤ -c 3 |x| 2 A , (6b)
[respectively, ∆V (x) ≤ -c 3 |x|

2 A ], (6c)
for suitable positive constants c1 , c2 and c3 , where |x| A denotes the distance of x from the set A. (iii) The closed attractor A in (3) is uniformly globally exponentially stable for the closed loop (1), ( 2). (iv) The closed loop (1), ( 2) is such that the sub-states

x i uniformly globally exponentially synchronize to the unique solution to the following initial value problem:

δx • = Ax • , x • (0) = 1 N N k=1
x k (0).

Proof: we first show a preliminary transformation, then we prove the theorem in four steps: (i) =⇒ (ii), (ii) =⇒ (iii), (iii) =⇒ (iv), and (iv) =⇒ (i).

Preliminary transformation. Let us define the extended state vector x = x 1 . . . x N and rewrite interconnection (1), (2) in the following compact form:

δx = (I N ⊗ A)x + (I N ⊗ B)u (8a) y = (I N ⊗ C)x (8b) u = -(L ⊗ C)x = -(I N ⊗ C)(L ⊗ I n )x, (8c)
where

I N ⊗ A ∈ R N n×N n , I N ⊗ B ∈ R N n×N , I N ⊗ C ∈ R N ×N n and L⊗I n ∈ R N n×N n .
Since matrix L is symmetric, there exists a unitary matrix T ∈ R N ×N (namely a matrix satisfying T T = I N ) that diagonalizes L. In particular, let us pick T such that:

Λ = T LT = diag (0, λ 1 , . . . , λ N -1 ) . (9)
Since the upper-left entry of Λ is zero, we may select T such that its first column corresponds to the eigenvector t 0 = 1 √ N 1 N associated to the zero eigenvalue λ 0 = 0 of L. Furthermore, it is easily checked that T ⊗ I n transforms L ⊗ I n into Λ ⊗ I n . Indeed, using the associative property of the Kronecker product we get

(T ⊗ I n ) (L ⊗ I n )(T ⊗ I n ) = (T LT ⊗ I n ) = Λ ⊗ I n .
Let us now introduce the similarity transformation x = (T ⊗ I n )x. Then dynamics (8a) reads:

δx = (T ⊗I n ) -1 (I N ⊗A)(T ⊗I n )x + (10a) +(T ⊗I n ) -1 (I N ⊗B)u y = (I N ⊗ C)(T ⊗ I n )x (10b) u = -(I N ⊗ C)(L ⊗ I n )(T ⊗ I n )x. (10c)
Substituting in [START_REF] Fax | Graph Laplacians and stabilization of vehicle formations[END_REF] the control law (10c) into (10a) and using the associative property of the Kronecker product we obtain:

δ x = Āx, (11)
where the state matrix A can be computed as:

Ā = (T -1 T ⊗A) -(T ⊗I n ) -1 (I N ⊗B)(I N ⊗C)(L⊗I n )(T ⊗I n ) = (I N ⊗ A) -(T -1 LT ⊗ BC) = (I N ⊗ A) -(Λ ⊗ BC) = (I N ⊗ A) -(I N ⊗ BC)(Λ ⊗ I n ), (12)
which has the following block diagonal structure:

Ā = diag (A, A 1 , . . . , A N -1 ) , (13)
by using the definitions in [START_REF] Chen | Rate adaptation and admission control for video transmission with subjective quality constraints[END_REF]. Proof of (i) =⇒ (ii). By assumption (5), we have that there exist matrices P k , k = 1, . . . , N -1, such that:

A k P k + P k A k = -I n , k = 1, . . . , N -1 , t ∈ R, or (14a) A k P k A k -P k = -I n , k = 1, . . . , N -1 , t ∈ Z. (14b)
Consider the block diagonal matrix P = diag (0, P 1 , . . . , P N -1 ) and define the Lyapunov function candidate:

V (x) = x (T ⊗ I n ) x P (T ⊗ I n ) x = N -1 k=1 x k P k xk . (15)
Then, from equations ( 11) and (14a) or [START_REF] Fax | Information flow and cooperative control of vehicle formations[END_REF] and (14b) it follows that:

V (x) = N -1 k=1 x k (P k A k + A k P k )x k = - N -1 k=1 x k xk , or ∆V (x) = N -1 k=1 x k (A k P k A k -P k )x k = - N -1 k=1 x k xk . (16)
To prove ( 6), we use Lemma 3 (which is postponed in the appendix), after noticing that matrix T introduced in the preliminary step of the proof satisfies the assumption of the lemma. Then we also observe that, using matrix ∆ = diag{0, 1, . . . , 1} defined in Lemma 1, we have:

N -1 k=1 x2 k = x (∆ ⊗ I n )x = ((T ⊗ I n )x) (∆ ⊗ I n )(T ⊗ I n )x = x (T ∆T ⊗ I n )x. (17)
Therefore, using [START_REF] Kim | Stabilization of multi-agent dynamical systems for cyclic pursuit behavior[END_REF], positive definiteness of P k , k = 1, . . . , N -1, definition [START_REF] Hong | Distributed observers design for leader-following control of multi-agent networks[END_REF] and Lemma 1, we obtain:

V (x) ≤ max h∈{1,...,N -1} λ max (P h ) p N -1 k=1 |x k | 2 = px (T ∆T ⊗ I n )x ≤ c 2 p |x| 2 A , V (x) ≥ min h∈{1,...,N -1} λ min (P h ) p N -1 k=1 |x k | 2 = px (T ∆T ⊗ I n )x ≥ c 1 p |x| 2 A . (18)
Thus (6a) is proven with c1 = c 1 p and c2 = c 2 p. Finally, using ( 16), ( 17) and Lemma 1 we get:

V (x) ≤ -x (T ∆T ⊗ I n )x ≤ -c 1 |x| A , (19)
which coincides with (6b) with c3 = c 1 . Note that, the same majoration holds for the discrete-time case, so that (6c) is satisfied.

Proof of (ii) =⇒ (iii). Based on (6), the uniform global exponential stability of A in (3) follows from standard Lyapunov results (see, e.g., the discrete-and continuous-time special cases of the hybrid results in [START_REF] Teel | Lyapunov-based sufficient conditions for exponential stability in hybrid systems[END_REF]Theorem1]).

Proof of (iii) =⇒ (iv). Consider the dynamics of the state

x • (t) := 1 N N k=1
x k (t) and note that from (1):

δx • (t) = 1 N N k=1 δx k (t) = A N k=1 x k (t) + B N k=1 u k (t) = Ax • (t) + B 1 N L =0 y = Ax • (t), (20)
where 1 N L = 0 due to well known properties of Laplacian matrices. Then x • evolves autonomously according to [START_REF] Cicalo | Cross-layer algorithms for distortion-fair scalable video delivery over OFDMA wireless systems[END_REF] and corresponds to the average of states x k . Since from (ii) =⇒ (iii) we know that states x k exponentially synchronize to some consensus, then they must synchronize to their average value that is x • .

Proof of (iv) =⇒ (i). We prove this step by contradiction.

Assume that one of matrices A k in ( 5) is not Schur-Cohn, and assume without loss of generality that it is A N -1 . Consider the coordinate system in ( 11) with [START_REF] Hara | Stability analysis of linear systems with generalized frequency variables and its applications to formation control[END_REF]. Then, from the block diagonal structure of Ā, since A N -1 is not Schur-Cohn, there exists a vector ω * ∈ R n (an eigenvector of one of the nonconverging natural modes) such that the solution to (11

) from x * (0) = 0 . . . 0 ω * corresponds to x * (t) = 0 . . . 0 x N (t)
, where xN (t) does not converge to zero. As a consequence, the function in [START_REF] Hong | Distributed observers design for leader-following control of multi-agent networks[END_REF] along this solution corresponds to:

V (x * (t)) = V ((T ⊗ I n )x * (t)) = x N (t)P N xN (t),
which, from linearity, remains bounded away from zero. Then, using the first inequality in [START_REF] Li | Content-aware distortionfair video streaming in congested networks[END_REF] we have that |x * (t)| A is bounded away from zero, namely solution x * (t) does not converge to the consensus set. In other words, the components of x * (t) do not asymptotically synchronize, which contradicts item (iv).

III. CONSENSUS IN QUALITY-FAIR VIDEO DELIVERY

A. Background

In this paper we analyze the model considered in [START_REF] Changuel | Control of distributed servers for quality-fair delivery of multiple video streams[END_REF], in which delivery of video streams is performed to several mobile users sharing the same wireless resource. Quality fairness is is targeted, i.e., the system aims at ensuring at each time instant the same utility value for all video streams.

The dynamics of the i-th video stream, i = 1, . . . , N , is described by the following set of equations (conveniently reported from [4, equation (22)]):

a i (j) + = a i (j) + δa i (j) (21a) a d i (j) + = a i (j) (21b) Φ i (j) + = Φ i (j) + ∆U dd i (j) -U dd i (j) (21c) Π b i (j) + = Π b i (j) + (B i (j) -B 0 ) (21d) R ed i (j) + = R 0 - K eb P +K eb I T (B i (j) -B 0 ) - K eb I T Π b i (j) (21e) R edd i (j) + = R ed i (j) (21f) U dd i (j) + = f (a d i (j), R ed i (j)) (21g) B + i (j) = B i (j) + [R edd i (j) -R 0 (21h) + (K t P + K t I )∆U dd i (j) -K t I Φ i (j)]T Ū dd (j) = 1 N N k=1 U dd k (j) (21i)
The discrete-time nonlinear state-space representation in [START_REF] Liu | Dynamic output feedback control for consensus of multi-agent systems: An h-∞; approach[END_REF] considers N mobile users, indexed by the subscript i, connected to the same base station (BS) and sharing wireless resources provided by the BS to get streamed videos delivered by N remote servers. Time is assumed to be slotted with a period T . Each video delivery chain is assumed to be controlled in a synchronous way, with video streams consisting of group of pictures (GoP) of the same duration T . Control is performed in a media-aware network element (MANE).

The rate-utility function of the j-th GoP of the i-th stream is modeled by a nonlinear function U i (j) = f (a i (j) , R) of the video encoding rate R, parametrized by a vector a i (j), which value depends on the video characteristics. The evolution of a i (j) is described by (21a), with δa i (j) representing some uncontrolled perturbation modeling the variations with time of the rate-utility characteristics. A total transmission rate R c is assumed to be shared by the users. The encoding rate target is evaluated within the MANE using an internal PI controller (controller K int ) aiming at regulating the buffer level B i of the i-th stream around some reference buffer level B 0 , see (21d) and (21e). K eb P and K eb I are the proportional and integral control parameters for the encoding rates. R 0 = R c /N is the average rate, which would be allocated in a rate-fair scenario. The draining rate of the i-th buffer within the MANE is controlled so as to minimize the discrepancy ∆U i (j) of the utility U i (j) of the i-th program with respect to the average utility given by (21i). For that purpose, an external PI controller (controller K ext ) with parameters K t P and K t I is involved: programs with a utility less than average are drained faster, leading to an increase of the encoding rate. A one-period forward and backward delay between the MANE and the server is considered to account for moderate queuing delays in the network. Provided that T is of the order of the second, this is a realistic upper bound. The delay operators account for these delays in [START_REF] Liu | Dynamic output feedback control for consensus of multi-agent systems: An h-∞; approach[END_REF]. In [START_REF] Changuel | Control of multiple remote servers for quality-fair delivery of multimedia contents[END_REF], [START_REF] Changuel | Control of distributed servers for quality-fair delivery of multiple video streams[END_REF] the four PI controller gains are heuristically selected to ensure the asymptotic convergence of the utilities U i (j) in (21g) to a common value Ū , namely:

lim j→+∞ U i (j) = Ū , ∀i = 1, . . . , N. (22)
Based on the results of Section II, we will provide below a systematic optimal selection of these parameters that ensures [START_REF] Moreau | Stability of continuous-time distributed consensus algorithms[END_REF], as established in Theorem 2 in the forthcoming Section III-B.

B. Two PI control loops

System (21) can be rearranged in order to highlight the different contributions of two PI controllers. The first one essentially rejecting the constant bias B 0 , and the second one rejecting the constant bias R 0 and inducing consensus of the utilities of the video streams. The first PI controller (denoted by K int in Figure 1) corresponds to an internal loop and is characterized by (21d) and (21e), rewritten as:

Π b+ i = Π b i + ∆B i (23a) K ext P K int z -2 f ai Ū dd ∆U dd i κ 2 ∆B i ∆ Rdd i U dd i R 0 Σ int Σ ext Σ 0 Fig. 1: Block Diagram of the controlled system κ 1 = k int I T Π b i + k int P T ∆B i , (23b)
where

Π b i is the controller state, ∆B i = B i -B 0 is the controller input and κ 1 = -∆R e i = -(R e i -R 0 )
is the controller output. The integral and proportional gains k int I and k int P are defined as:

k int I = K eb I , k int P = K eb P + K eb I . (24)
The second PI controller (denoted by K ext in Figure 1) is characterized by (21c) and (21h), rewritten as:

Φ s+ i = Φ s i + ∆U dd i σ (25a) κ 2 = k ext I Φ s i + k ext P σ ∆U dd i ( 25b
)
∆U dd i = 1 N N k=1 U dd k -U dd i (25c)
where σ > 0 is a normalizing constant, Φ s i = Φi σ is the controller state vector, ∆U dd i is the controller input, and κ 2 is the controller output. The integral and proportional gains k ext I and k ext P are defined as:

k ext I = σK t I , k ext P = σ(K t P + K t I ), (26)
With this notation, (21e), (21f) and (21h) become:

∆R ed+ i = ∆R e i = -κ 1 (27a) ∆R edd+ i = ∆R ed i (27b) ∆B + i = ∆B i + T (∆R edd i -κ 2 ). (27c)
According to [START_REF] Moreau | Stability of multiagent systems with time-dependent communication links[END_REF], [START_REF] Ren | Consensus strategies for cooperative control of vehicle formations[END_REF] and as represented in Figure 1, controller K int performs a delayed negative feedback action over the plant through the delayed output ∆R dd i .

C. The system seen as a consensus feedback

Let Σ ext = (A ext , B ext , C ext , D ext ) denote the state-space representation for controller K ext (i.e., the system with input variable ∆U dd i and output variable κ 2 ), and Σ int = (A int , B int , C int , D int ) denote the state-space representation for the feedback loop that includes the controller K int (i.e., the system with input variable κ 2 and output variable ∆R edd i ). Then, using [START_REF] Moreau | Stability of multiagent systems with time-dependent communication links[END_REF] and [START_REF] Ren | Consensus strategies for cooperative control of vehicle formations[END_REF] for Σ int and (25) for Σ ext , one may represent the dynamics of the i-th video stream using the states x int and x ext defined as:

x int = ∆Bi T Πi T -∆R ed i -∆R edd i , x ext = Φ i . ( 28
)
With this selection, the state-space matrices of the subsystems are given by:

A ext B ext C ext D ext = 1 1 σ k ext I k ext P σ , (29)
A int B int C int D int =      1 0 0 1 -1 1 1 0 0 0 -k int P -k int I 0 0 0 0 0 -1 0 0 0 0 0 1 0      . ( 30
) According to Figure 1, one can then represent the inner dynamics of each video stream, represented by Σ 0 in Figure 1, as the cascaded interconnection of Σ ext and Σ int , establishing the linear relation from ∆U dd i to ∆R dd i + R 0 = R dd i , whose state-space representation Σ 0 = (A 0 , B 0 , C 0 , D 0 ) is such that the state matrix A 0 is lower-triangular. Actually, given the state vector x = x ext x int , the input variable ∆U dd i and the output variable ∆R dd i we have:

A 0 B 0 C 0 D 0 =   A ext 0 B ext B int C ext A int B int D ext 0 C int 0   . ( 31
)
Due to its lower block triangular structure the eigenvalues of A 0 are the union of the eigenvalues of A int and A ext . Then the overall system dynamics is influenced by the separate actions of the two subsystems Σ int and Σ ext . In particular Σ int performs an internal stabilizing action of each stream dynamics, and Σ ext performs the external synchronization among the streams over the network.

D. Main consensus theorem

The coupling among the different video streams arises from the action of the average utility Ū dd in (21i), acting as an input to each video stream dynamics, where the utility U dd i of each stream is a nonlinear function of the state a i in (21a) and (21b). In particular, it is easily shown that (21g) leads to:

U dd i = f (a dd i , R edd i ) = f (a dd i , ∆R edd i + R 0 ), (32)
so that U dd i can be seen as a nonlinear time-varying output of system Σ 0 in [START_REF] Rudin | Functional Analysis. Mathematics series[END_REF]. In this paper we make the following strong assumption, so that a linear time-invariant analysis of the consensus algorithm can be performed.

Assumption 1: For each i = 1, . . . , N , the input δa i in (21a) is zero, so that a i is constant for each i. Moreover there exist scalars h i , i = 1, . . . , N and a scalar K f > 0 such that:

U dd i = f (a dd i , R edd i ) = h i + K f R edd i (33) = h i + K f R 0 + K f ∆R edd i , ∀i = 1, . . . , N.
Intuitively speaking, K f translates the variation of utility provided by a variation of the video encoding rate. Based on Assumption 1 and on the presence of the integral action of controller K ext , we may perform a coordinate change to compensate for the action of the constant disturbance h i + K f R 0 , so that the overall system can be written as an output feedback network interconnection of N identical linear systems:

x

+ i = A 0 x i + B 0 ∆U dd i U dd i = K f C 0 x i ∀i = 1, . . . , N. (34)
In particular, using (25c), each input ∆U dd i can be expressed, for each i = 1, . . . , N , as:

∆U dd i = Ū dd -U dd i = 1 N j =i U dd j -1 - 1 N U dd i . ( 35
)
Define now the vectors U dd = U dd . Then [START_REF] Sepulchre | Consensus on nonlinear spaces[END_REF], for i = 1, . . . , N , can be rewritten in the compact form:

∆U dd = -    1-1 N -1 N ... ... -1 N -1 N 1-1 N ... ... -1 N . . . . . . . . . . . . -1 N -1 N ... 1-1 N    U dd = -LU dd , ( 36
)
where L is the N × N Laplacian matrix associated with the network. The Laplacian matrix resumes the information exchanged by the subsystems. Notice that the graph related to the network described by matrix L defined in (36) is fully connected, i.e., every vertex has an edge to every other vertex [START_REF] Fax | Information flow and cooperative control of vehicle formations[END_REF]. Combining [START_REF] Seo | Consensus of high-order linear systems using dynamic output feedback compensator: low gain approach[END_REF] and [START_REF] Sepulchre | Consensus on nonlinear spaces[END_REF], we obtain the compact form for the overall system:

x + = (I N ⊗ A 0 )x + (I N ⊗ B 0 )(-Ly) (37a) y = U dd = K f (I N ⊗ C 0 )x, ( 37b
)
where y is the output representing the N utilities and x = x 1 . . . x N is the overall state of the interconnected systems.

Then the following theorem can be stated. Theorem 2: Under Assumption 1, the following statements are equivalent:

(i) For any initial condition, all utilities U i , i = 1, . . . , N of model ( 23), ( 25), ( 27), [START_REF] Scardovi | Synchronization in networks of identical linear systems[END_REF] converge to the same value, i.e., condition ( 22) is satisfied. (ii) Given any solution to [START_REF] Slotine | A study of synchronization and group cooperation using partial contraction theory[END_REF], there exists Ū ∈ R such that lim j→+∞ y i (j) = Ū , ∀i = 1, . . . , N . (iii) The consensus set A in (3) is uniformly globally exponentially stable for dynamics [START_REF] Slotine | A study of synchronization and group cooperation using partial contraction theory[END_REF] and matrix A int is Schur-Cohn. (iv) Matrix A int and matrix

A f = A 0 -K f N N -1 B 0 C 0 (38)
are both Schur-Cohn.

Proof: we will prove the theorem according to the following steps: (i) ⇔ (ii), (iii) ⇔ (iv), (iv) ⇒ (i) and (i) ⇒ (iv). Proof of (i) ⇔ (ii). This equivalence follows from the fact that, due to relations (34)- [START_REF] Seuret | Consensus under communication delays[END_REF], and from the definitions in ( 29)- [START_REF] Scardovi | Synchronization in networks of identical linear systems[END_REF], model [START_REF] Slotine | A study of synchronization and group cooperation using partial contraction theory[END_REF] coincides with the closed loop ( 23), ( 25), ( 27), [START_REF] Scardovi | Synchronization in networks of identical linear systems[END_REF]. Proof of (iii) ⇔ (iv). Applying the equivalence between items (i) and (iii) of Theorem 1 when focusing on system [START_REF] Slotine | A study of synchronization and group cooperation using partial contraction theory[END_REF], item (iii) of Theorem 2 is equivalent to having that all eigenvalues λ k of matrix L in [START_REF] Seuret | Consensus under communication delays[END_REF], except for that one related to the eigenvector 1 N , are such that A 0 -λ k K f B 0 C 0 is Schur-Cohn. Since all the eigenvalues (except that one equals to zero) of the Laplacian matrix L are equal to N N -1 , the result trivially follows.

Proof of (iv) ⇒ (i). Similar to the previous step, this implication follows from item (iv) of Theorem 1 after noticing that system x + • = Ax • corresponds to system Σ 0 , namely A = A 0 , where A 0 is given in [START_REF] Rudin | Functional Analysis. Mathematics series[END_REF]. Since A int is Schur-Cohn by assumption, then due to its block triangular structure, matrix A 0 has a single eigenvalue at zero and all solutions to (7) converge to a constant, thereby proving item (i) of Theorem 2. Proof of (i) ⇒ (iv). We prove this by contradiction. Assume that item (iv) does not hold. Then either A f is not Schur-Cohn, which implies from Theorem 1 that consensus is not achieved for some initial conditions (thereby proving that (i) does not hold), or A f is Schur-Cohn and A int is not Schur-Cohn. In this case, Theorem 1 applies because A f is Schur-Cohn and all solutions exponentially synchronize to a solution to [START_REF] Cicalo | Cross-layer algorithms for distortion-fair scalable video delivery over OFDMA wireless systems[END_REF] with A = A 0 as in [START_REF] Rudin | Functional Analysis. Mathematics series[END_REF]. Then two cases may occur: a) A int has at least one eigenvalue with magnitude larger than 1 or at least one eigenvalue on the unit circle with multiplicity larger than 1: in this case some solutions synchronize to a diverging evolution, thus item (i) does not hold; b) A int has at least one eigenvalue with magnitude 1 on the unit disk. If that eigenvalue is at 1, then due to the triangular structure, matrix A 0 has two eigenvalues in 1 (the other one coming from A ext ) and again some solutions synchronize to a diverging evolution. If that eigenvalue is anywhere else in the unit circle, then it generates a revolving non-constant mode and some solutions synchronize to a non-convergent oscillatory evolution. In both cases a) and b), item (i) does not hold and the proof is completed.

IV. CONTROL DESIGN

In this section we address the problem of finding suitable gains K ext , K int in order to guarantee i) the asymptotic stability of matrices A f and A int in item (iv) of Theorem 2 and ii) the optimization of the system performance in terms of the convergence rate of A int and A ext . The basic idea consists in designing the PI controller through a numerical technique based on an iterative LMI approach. The proposed algorithm allows addressing general static output feedback design problems, and can be viewed as an alternative approach to coordinate descent algorithms [START_REF] Arzelier | An iterative method for mixed h 2 h∞ synthesis via static output-feedback[END_REF], [START_REF] Ebihara | S-variable approach to lmibased robust control[END_REF]. The gain selection consists in a two-steps optimization process in which first the controller K int is designed in order to maximize the convergence rate of A int , and once K int is fixed, the same procedure will be applied to the selection of K ext in order to maximize the convergence rate of A f . Let us first note that, after a suitable permutation of the state variables, matrices A int in [START_REF] Ren | Information consensus in multivehicle cooperative control[END_REF] and A f in [START_REF] Stan | Analysis of interconnected oscillators by dissipativity theory[END_REF]

A int = A 1 -B 1 K int [I 0] , (39)
A f = A 2 -B 2 K ext [I 0] , (40)
where we have defined:

A 1 B 1 C 1 D 1 =     1 0 0 -1 0 1 1 0 0 0 0 0 0 0 1 0 0 -1 0 0 1 0 0 0 0 0 1 0 0 0     , (41)
A 2 B 2 C 2 D 2 =       1 1 0 0 0 0 0 -1 0 0 1 1 0 0 1 0 1 0 0 0 -k int I 0 -k int P 0 0 0 0 -1 0 0 1 0 0 0 0 0 0 1 0 0 0 0       . ( 42
)
A possible solution to this is given in Algorithm 1. The core idea is to alternate between two main steps, each of them requiring the solution of a quasi-convex optimization problem, i.e., a generalized eigenvalue problem (GEVP) based on LMI and bisection, where the controller gain K and a multiplier G are alternated as optimization variables. Algorithm 1 joins several useful properties that make it a promising tool for computing suboptimal selections of the static output feedback gain. Clearly, there is no guarantee of optimality, as the static output feedback problem is well known to be a challenging and nonconvex one. Some useful properties of Algorithm 1 are stated and proven next.

Proposition 1: The following statements hold: (i) Initialization and termination: Given any input (A, B, [I 0]) and δ > 0, the Initialization step in Algorithm 1 gives an admissible pair, and the algorithm terminates after a finite number of steps. (ii) Feasibility: Given any admissible pair (α L1 , α H1 ) from

Step 1, the pair (α L2 , α H2 ) obtained from the subsequent

Step 2 always satisfies α L2 ≥ α L1 , and viceversa. (iii) Guarantees: Any solution

(K out , α out ) resulting from Algorithm 1 satisfies σ (A -BK out [I 0]) ⊆ C ≤β , where β = √ 1 -α out .
In particular, if α out > 0, then the gain selection K out is a stabilizing output feedback gain for the triple (A, B, [I 0]), and β is the corresponding convergence rate. Proof: Proof of (i). First, we prove that (α 0 , 1.1) is an admissible pair in the sense clarified in the initialization step. Trivially, ( 44) is infeasible with α > 1, because the upper-left entry is positive. To show that ( 44) is feasible with α = α L = σ 0 as in [START_REF] Wang | Consensus of multi-agent linear dynamic systems[END_REF], select G 11 = I, G 22 = I, G 21 = 0, X 1 = 0, and W = I so that, applying a Schur complement, ( 44) is feasible if:

(σ 0 -1) I + AA ≤ 0, (47)
Algorithm 1 Convergence rate α and controller K Input: Matrices A, B, C = [I 0], and a tolerance δ > 0.

Initialization: Set M = 0 and initialize the pair (α L , α U ) = (σ 0 , 1.1), where, using σ(A) to denote the maximum singular value of A, we select:

σ 0 = 1 -σ2 (A). (43)
Pair (α L , α U ) is admissible for [START_REF] Wang | Finite-time consensus problems for networks of dynamic agents[END_REF], in the sense that ( 44) is feasible with α = α L and infeasible with α = α U .

Iteration

Step 1: Given M and pair (α L , α H ) from the previous step, solve, using bisection with tolerance δ > 0 , the GEVP: max

W,G11,G21,G22,X1,α α s.t. -W + αW AG -BX -G -G + W ≤ 0, (44)
where W = W > 0 and matrices, G and X have the following structure (see, for example, [START_REF] Pipeleers | Extended lmi characterizations for stability and performance of linear systems[END_REF] for details on the use of multipliers):

G = G 11 G 11 M G 21 G 22 , X = X 1 [I M ] . (45)
In particular, determine an admissible pair (α L , α H ) such that α H -α L ≤ δ. Pick the (sub)optimal solution Ḡ11 , X1 corresponding to α L , and set K = Ḡ-1 11 X1 for the next step.

Step 2: Given K and pair (α L , α U ) from the previous step, set Ā = A-B KC, and solve, using bisection with tolerance δ > 0 , the GEVP:

max α,W =W >0 α (46)
s.t. ĀW Ā -W ≤ -αW.

In particular, determine an admissible pair (α L , α H ) such that α H -α L ≤ δ. Pick the (sub)optimal solution W = W11 W12 W21 W22 (where W has the partition induced by G), corresponding to α L and set M = W -1 11 W12 for the next step. until α L does not increase more than δ over three consecutive steps. Output:

K out = K and α out = α L .
which is clearly ensured if σ 0 -1 + σ2 (A) ≤ 0. We now prove that the algorithm always terminates in a finite number of steps. Let α j L1 denote the value of α L at the j-th iteration of Step 1. From item (ii) of Proposition 1 the sequence α j L1 , j ∈ N, is non decreasing and upper bounded by α = 1, thus it is convergent, i.e., given δ > 0 there exists an index j ∈ N such that α j+1 L1 -α j L1 ≤ δ. Proof of (ii). [From Step 1 to Step 2]. By substituting the solution α L1 , K obtained from Step 1 in [START_REF] Wang | Finite-time consensus problems for networks of dynamic agents[END_REF] we get that:

-W + α L1 W (A -B KC)G -G -G + W ≤ 0 (48)
has a feasible solution. By applying Finsler's Lemma, feasi-bility of ( 48) is equivalent to feasibility of:

(A -B KC)W (A -B KC) -W ≤ -α L1 W. (49)
Comparing ( 49) with [START_REF] Wieland | An internal model principle is necessary and sufficient for linear output synchronization[END_REF], it follows that the subsequent solution α L2 to Step 2 satisfies α L2 ≥ α L1 .

[From Step 2 to Step 1]. Substitute the solution α L2 , M obtained from Step 2 in [START_REF] Wieland | An internal model principle is necessary and sufficient for linear output synchronization[END_REF] and perform a Schur complement to get:

-W + α L2 W (A -BKC)W -W -W + W ≤ 0, (50)
which corresponds to [START_REF] Wang | Finite-time consensus problems for networks of dynamic agents[END_REF] with W = G. It follows that the subsequent solution α L1 to Step 1 satisfies α L1 ≥ α L2 .

Proof of (iii). From linear systems theory [START_REF] Hespanha | Linear Systems Theory[END_REF], we get that both solutions at Step 1 and Step 2 provide a certificate that matrix Ā = A -BK out C has a spectral radius smaller than α out . Remark 1: There is no loss of generality in considering systems is the form (A, B, [I 0]) in Algorithm 1. For a system in a general form (A, B, C), where matrix C is fullrow rank, there always exists a nonsingular matrix T such that CT -1 = [I 0]. Using T as a similarity transformation we obtain

T -1 AT, T -1B, CT = Ā, B, [I 0] .
To demonstrate the effectiveness and the convergence of the proposed algorithm, the outer and inner loop gains K int and K ext of the application discussed on Section III are designed using the general procedure in Algorithm 1, with tolerance δ = 10 -8 , and with the selections in ( 41) and ( 42), respectively. Figure 2a shows that after 32 iterations, the rate α L related to the selection of K int (see [START_REF] Stefanov | Separable Programming: Theory and Methods[END_REF], ( 41)) corresponds to 0.37789, with K int = [0.19256 0.012915]. Figure 2b shows similar results for the selection of K ext : after 33 iterations the value of α L is 0.1165 with K ext = [0.17645 0.65801]. In Table I the results of the algorithm in comparison with [START_REF] Col | A linear consensus approach to quality-fair video delivery[END_REF] are summarized. In our preliminary work [START_REF] Col | A linear consensus approach to quality-fair video delivery[END_REF] we used a graphical method to solve the disign problem for K int and K ext with the same performance goal as that one of Algorithm 1. Table I shows that Algorithm 1 provides similar results to those of [START_REF] Col | A linear consensus approach to quality-fair video delivery[END_REF] by using a systematic approach, which can be generalized to systems of any order, contrarily to the graphical method in [START_REF] Col | A linear consensus approach to quality-fair video delivery[END_REF]. [START_REF] Col | A linear consensus approach to quality-fair video delivery[END_REF] and those obtained from Algorithm 1.

V. SIMULATION RESULTS In this section we provide simulation results for the control scheme [START_REF] Liu | Dynamic output feedback control for consensus of multi-agent systems: An h-∞; approach[END_REF] characterized in Theorem 2, with the gain selection in [START_REF] Olfati-Saber | Consensus and cooperation in networked multi-agent systems[END_REF], [START_REF] Pipeleers | Extended lmi characterizations for stability and performance of linear systems[END_REF] and Table I. This selection guarantees item (iv) (therefore all other items) of Theorem 2, as [START_REF] Cho | Utility max-min flow control using sloperestricted utility functions[END_REF]. The frame rate is F = 30 frames/s. GoPs of 10 frames are considered, thus the GoP duration is T = 0.33 s. The considered utility U i is the Peak Signal-to-Noise Ratio (PSNR). To tune the controllers, the rate-utility characteristics of each GoP is estimated as described in [START_REF] Changuel | Control of multiple remote servers for quality-fair delivery of multimedia contents[END_REF], [START_REF] Changuel | Control of distributed servers for quality-fair delivery of multiple video streams[END_REF]. The control is assumed to be performed within the MANE, closely located to the BS to which the clients are connected. A Matlab simulation of the behavior of the servers, the network, the MANE, and the clients is performed. The forward and backward propagation and queuing delays between the MANE and the servers are taken as constant and equal to T . The packets delivered by the MANE to the BS and to the clients are assumed to be well received thanks to retransmission at the MAC layer, which is not modeled here. During the control of the streaming system, the rate-utility characteristics are not available at the MANE. Only the utility of the encoded packets it receives are used. They may be tagged, e.g., at the RTP layer of the protocol stack. The MANE adjusts the transmission rate of each stream and provides an encoding rate target to the individual servers, which are then responsible of meeting this target by video encoding, transcoding, or bit-rate switching. Model [START_REF] Liu | Dynamic output feedback control for consensus of multi-agent systems: An h-∞; approach[END_REF] developed in [START_REF] Changuel | Control of multiple remote servers for quality-fair delivery of multimedia contents[END_REF] has been simulated setting the parameters as follows. We have chosen B 0 = 1200 kb to tolerate significant variations of the buffering delay, and the channel rate is R 0 = 4000 kb/s. Two triplets of simulations are presented in Figures 3 and4, corresponding to different selection of N = 4 video streams out of the six ones described above. In particular, each figure shows comparatively the evolution of the utilities U i , namely, as commented before, the PSNR of the streams, for three selections of the control parameters K eb I , K eb P , K t I , and K t P in [START_REF] Liu | Dynamic output feedback control for consensus of multi-agent systems: An h-∞; approach[END_REF], reported in Table II and arising from: (a) Subfigures (a) (left). Tuning based on Theorem 2 and Algorithm 1. The value of the parameter K f introduced in (33) has to be selected. To this end, the time and ensemble average of the rate-PSNR characteristics for the four first streams have been evaluated at different constant encoding rates ranging from R e = 250kb/s to R e = 2Mb/s. The resulting values of K f range from k f = 0.02 dB/kb/s to k f = 0.0025 dB/kb/s. To avoid too aggressive variations of the video encoding rate and increase robustness of the system, K f = 0.02 dB/kb/s has been selected. Considering N = 4, the PI gains in ( 21) can be chosen using the values in Table I within ( 24) and ( 26), with σ = K f N/(N -1) = 0.0267. (b) Subfigures (b) (middle). Tuning based on [START_REF] Changuel | Control of multiple remote servers for quality-fair delivery of multimedia contents[END_REF]. The parameters proposed in [START_REF] Changuel | Control of multiple remote servers for quality-fair delivery of multimedia contents[END_REF] reported in Rate-Fair (TRF) scheme. In the TRF scheme, no external controller is applied, so that the transmission rate is always the same for all programs, while the internal controller gains are set as in [START_REF] Changuel | Control of multiple remote servers for quality-fair delivery of multimedia contents[END_REF]. Also in this case Theorem 2 anticipates lack of consensus, because matrix A f has poles at the limit of stability (indeed the scheme corresponds to L = 0 N,N , with only zero eigenvalues). A straightforward extension of Theorem 1 predicts lack of convergence but also lack of convergence of the utilities. The first four video streams, which have been used for the tuning of K f , are considered in the experiments reported in Figure 3, showing the utilities of the four users. Figure 4 shows the utilities in another simulation using the last four videos. With the aforementioned schemes, the average absolute value of the difference of the PSNR of each stream and the average PSNR is evaluated as follows:

∆U = 1 M N M j=1 N k=1 U k (j) -U (j) . (51)
Table III captures the results of the simulations presented in Figures 3 and4. We notice that the presented scheme gives better results in terms of average utility in comparison with the other control strategies. In particular, since the scheme in [START_REF] Changuel | Control of multiple remote servers for quality-fair delivery of multimedia contents[END_REF] does not stabilize the consensus set, it is not surprising that the utilities obtained with this scheme diverge when sending the last four videos (see Figure 4b), and we were nopt able to evaluate the corresponding average utility.

VI. CONCLUDING REMARKS

Consensus for multi-agent systems has been characterized in the case of N identical (continuous-or discrete-time) systems connected through a network of fixed topology. Necessary and sufficient conditions have been proven to ensure uniform global exponential stability of the state consensus set. The results have been applied in the context of the quality-fair delivery of videos and general LMI-based iterative procedure addressing design of static output feedback linear stabilizers has been proposed to design (sub)-optimally the PI controller gains. Future research directions include the extension of the methodology to obtain alternative performance guarantees. Furthermore, it might be useful to adapt the results to the case where the input of the system is limited in magnitude (saturation).

Moreover, y ∈ A satisfies (52) if and only if x ∈ N A (y), where:

N A (y) = {n ∈ R ν : n -y, y -a ≥ 0 ∀a ∈ A} (53)
is the normal cone to A at y, and y is the orthogonal projection of x onto A (see [START_REF] Rockafellar | Variational Analysis[END_REF]). Proof: we only prove the equivalence among ( 52) and (53) because the existence and uniqueness of y is already proven in [START_REF] Rudin | Functional Analysis. Mathematics series[END_REF]Theorem 12.3]. Proof of (53) ⇒ (52). If x ∈ N A (y) then, ∀a ∈ A we have: (a) Control parameters proposed here (Table I). (a) Control parameters proposed here (Table I). which, dividing by η, implies: 2 x -y, y -a + η |y -a| 2 ≥ 0.

|x -a| 2 = |x -y + y -a| 2 = |x -y| 2 + |y -a| 2 + 2 x -y, y -a ≥ |x -y| 2 .
Taking the limit as η → 0, the statement is proven.

Using Lemma 2 we can prove the following: Then, according to Lemma 2, the proof is completed if x ∈ N A (y). To prove this fact, first note that, since A is a linear subspace, for any pair of vectors y, a ∈ A, we have b := y -a ∈ A, so that it is enough to show:

x -y, b ≥ 0, ∀b ∈ A.

Relation (55) can be established by first noticing that b ∈ A implies that there exists b ∈ R n such that b = 1 N ⊗ b, and then computing:

x -y, b = x -1 N ⊗ x, 1 N ⊗ b = 1 N ⊗ b, x -1 N ⊗ x = (1 N ⊗ b) x -1 N ⊗ 1 N (1 N ⊗ I n )x = 1 N (1 N ⊗ b ) N I N n -1 N ⊗ 1 N ⊗ I n x = 1 N (1 N ⊗ b ) N I N -1 N ⊗ 1 N ⊗ I n x = 1 N   1 N N I N -1 N ⊗ 1 N =0 ⊗ b    x = 0
which completes the proof. x, (61)

Proof of

which implies |x ⊗ 1 n -x| 2 = (x ⊗ 1 n -x) (x ⊗ 1 n -x) = x T 2 T 2 x ≤ k 2 |x| 2 ,
where k 2 is the maximum singular value of T 2 T 2 .

1 .

 1 . . U dd N and ∆U dd = ∆U dd 1 . . . ∆U dd N

  can be conveniently rewritten as a function of the controllers K int = k int P

Fig. 2 :

 2 Fig. 2: Gain Selection via Algorithm 1. Evolution of α L through the iterations: (a) A int and (b) A f .

2 .

 2 (c) Subfigures (c) (right). Tuning based on a Transmission

2 :

 2 1, let us introduce some useful results. The following Lemma is based on [39, Theorem 1.10].Lemma Given a closed, convex set A ⊂ R ν and any vector x ∈ R ν , there exists a unique point y ∈ A satisfying:|x -y| = |x| A := min a∈A |x -a| .

  Control parameters taken from[START_REF] Changuel | Control of multiple remote servers for quality-fair delivery of multimedia contents[END_REF].

  Transmission rate fair streaming.

Fig. 3 :

 3 Fig. 3: PSNR of Progs 1 to 4, comparison between different control schemes. .

  Control parameters taken from[START_REF] Changuel | Control of multiple remote servers for quality-fair delivery of multimedia contents[END_REF].

  Transmission rate fair streaming.

Fig. 4 : 2 = 2 =

 422 Fig. 4: PSNR of Progs 3 to 6, comparison between different control schemes.

Lemma 3 : 1 N

 31 For any pair of positive integers n, N , given set A in (3), we have for all x ∈ R N n : ⊗I n )x, (54) where x k ∈ R n and x ∈ R n is the (vector) average of the (vector) components of x ∈ R N n .Proof: let us select y = 1 N ⊗ x ∈ R nN , so that |x -y| 2 = N k=1 |x k -x| 2 .

Lemma 1 :(x 1 - 2 =

 112 based on Lemma 3, we can now prove Lemma 1. Since matrix ∆ has a zero in the upper left entry and ones in the remaining diagonal entries, we can write:T ∆T = T T(56)whereT ∈ R N ×(N -1) , composed by the last N -1 columns of T , satisfies T 1 N = 0 and has N -1 independent columns. Therefore, Im T ⊂ (1 N ) ⊥ . As a consequence Im Im T , and there exists Σ invertible suchthat (Σ T ⊗ I n )x = x := x1-x2 . . . x1-x N ∈ R (N -1)n , wherex clearly satisfies N k=2 x k ) 2 = |x| 2 . From relation (56)consider now the quadratic form:x (T ∆T ⊗ I n )x = (M ⊗ I n )x, where x = x 1 -x 2 . . . x 1 -x N . Then noticing that λ min (M ⊗ I n ) = λ min (M ) = c 1 and λ max (M ⊗ I n ) = λ max (M ) = c 2we obtain the inner inequalities in (3). To complete the proof we need to show the outer inequalities in (3). To this end, it is sufficient to show that there exist positive scalars k 1 and k 2 such that for any pair n, N and any x ∈ R N n : result follows from Lemma 3. To show (57) we first observe thatN k=1 |x -x k | 2 = |x ⊗ 1 n -x| 2and then the straightforward relation:x x = (x ⊗ 1 n -x) T 1 T 1 (x ⊗ 1 n -x) ≤ k -1 1 |x ⊗ 1 n -x| , where k -11 is the maximum singular value of T 1 T 1 . Similarly we have:1N [-I n -I n . . . -I n ] x -1)I n -I n . . . -I n ] x = = x -x 1 + N N (x 1 -x 2 ) = x -x 2(60)Using similar reasonings, one gets:(x ⊗ 1 n -x)

TABLE I :

 I Comparision between the PI controller gains K int and K ext provided in

Table I

 I

	, six video streams 1
	of different types have been encoded during 60 s with x.264
	[47] in 4CIF (704 × 576) format at various bit rates. The
	programs are Interview (Prog 1), Sport (Prog 2), Big Buck
	Bunny (Prog 3), Nature Documentary (Prog 4), Video Clip
	(Prog 5), and an extract of Spiderman (Prog

  Table II that correspond to the selection K int = [0.152 0.002], K ext = [0.418 0.5944]. With this selection one can easily verify that matrix A int is Schur-Cohn, but A f has unstable eigenvalues. Then Theorem 2 anticipates lack consensus. This is confirmed by the middle plots of Figures 3 and 4

TABLE II :

 II Comparison between the gain tuning for the control schemes.

		Algorithm 1	TRF scheme	Scheme [3]
	Progs 1-4	2.66	4.48	2.96
	Progs 3-6	3.55	3.98	-

TABLE III :

 III Comparision of the average utility ∆U obtained with different control schemes (the values are in dB).

http://www.youtube.com/watch?v=l2Y5nIbvHLs, =G63TOHluqno, =YE7VzlLtp-4, =NNGDj9IeAuI, =rYEDA3JcQqw, =SYFFVxcRDbQ.

Note that the gains in[START_REF] Changuel | Control of multiple remote servers for quality-fair delivery of multimedia contents[END_REF] had been heuristically tuned without any formal guarantee of convergence.

M. Kieffer is partly supported by the Institut Universitaire de France. Parts of this work were previously presented at IEEE CDC 2014. Work supported by the ANR project LimICoS contract number 12 BS03 005 01.