Van Thiem Pham

Nadhir Messai
email: nadhir.messai@univ-reims.fr

Hoa Dinh
email: hoa.nd@i2cner.kyushu-u.ac.jp

Nguyen

Noureddine Manamanni
email: noureddine.manamanni@univ-reims.fr

Robust Formation Control Under State Constraints of Multi-Agent Systems in Clustered Networks

Keywords: Clustered network, state constraints, hybrid communication, formation control, linear matrix inequalities (LMIs)

This paper studies the formation control problem in clustered network systems composing of linear agents that are subjected to state constraints. In each cluster, there exists an agent called a leader who can communicate with other leaders outside of its cluster at some specific discrete instants. Moreover, the continuous-time communication structure in each cluster is represented by a fixed and undirected graph. A robust formation control protocol is proposed to deal with the hybrid communication described above and the constraints on states of agents. It is next shown that the hybrid robust formation control design for clustered multi-agent networks can be indirectly solved through the robust stabilization design of an equivalent system obtained by matrix theory and algebraic graph theory. Then, a robust controller is designed for the initial clustered network system in terms of linear matrix inequalities. Finally, a formation design for unmanned aerial vehicles is carried out and simulated to illustrate the effectiveness of the proposed hybrid formation control design method.

Introduction

Multi-agent systems (MASs) and their cooperative control problems have been extensively investigated in the past two decades, motivated by many practical applications such as unmanned aerial vehicles (UAVs), wireless sensor networks, autonomous underwater vehicles (AUVs), transportation networks, satellite networks, etc. Formation is one of the most interesting problems and intensively investigated issue in MASs, because of both several theoretical challenges and a wide range of potential applications [START_REF] Zhao | Distributed finite-time tracking control for multi-agent systems: An observer-based approach[END_REF][START_REF] Liu | Finite-time formation control for linear multi-agent systems: A motion planning approach[END_REF][START_REF] Guo | Local control strategy for moving-target-enclosing under dynamically changing network topology[END_REF][START_REF] Dong | Output containment control for swarm systems with general linear dynamics: A dynamic output feedback approach[END_REF].

Currently, consensus-based control approaches are widely employed in solving the formation problem of MASs. The firstorder MASs was studied in [START_REF] Ren | Multi-vehicle consensus with a time-varying reference state[END_REF]. It was proved that they could achieve formation if the directed interaction topology contains a directed spanning tree. Extensions of consensus-based formation algorithms to second-order with linear dynamics under the fixed or the time-varying interaction graphs were also investigated in [START_REF] Ren | Distributed multi-vehicle coordinated control via local information exchange[END_REF]. In [START_REF] Oh | Formation Control and Network Localization via Orientation Alignment[END_REF], a formation control law based on local measurement of relative-positions was proposed for first-order MASs. In another research [START_REF] Dong | Timevarying formation control for unmanned aerial vehicles: Theories and applications[END_REF], based on consensus approaches, necessary and sufficient conditions were derived to deal with the time-varying formations for second-order UAV swarm systems. Another direction is to study the formation of secondorder MASs with time delays [START_REF] Cheng | Formation control of multi-agent systems with heterogeneous communication delays[END_REF]. Accordingly, the sufficient conditions were proposed for MASs to achieve desired stationary and moving formations. Next, the formation tracking control problems [START_REF] Han | Formation tracking control for time-delayed multi-agent systems with second-order dynamics[END_REF], which uses neighboring relative state, and position information for second-order MASs with time-varying delays, were considered. Moreover, the formation tracking problems were also investigated in [START_REF] Yang | Fault-tolerant Consensus of Leader-following Multi-agent Systems Based on Distributed Fault Estimation Observer[END_REF], where the agents' local coordinate systems are applied such that the centroid of the controlled formation tracks a given trajectory. Recently, some new studies concern the influence of environments on wireless communication [START_REF] Li | Robust Distributed Consensus Control of Uncertain Multi-Agents Interacted by Eigenvalue-Bounded Topologies[END_REF][START_REF] Gao | Distributed H ∞ Control of Platoon Interacted by Switching and Undirected Topology[END_REF]. This leads to the degradation of the performance of wireless communication, which causes the interaction topology among agents cannot hold for a long time.

Other significant and realistic issues have been encountered such as the constraints on the agent's inputs, states, or relative states because of the physical limitations of agents. This includes, for example, the formation of vehicles with limited speeds and limited working space, smart buildings energy control with constraints on temperature and humidity in specific ranges and so on. Recently, some studies have considered the cooperative control of MASs under the constraints on agent's inputs, states, or relative states [START_REF] Nedi | Constrained Consensus and Optimization in Multi-Agent Networks[END_REF][START_REF] Wei | Consensus problems for linear time-invariant multi-agent systems with saturation constraints[END_REF][START_REF] Mo | Distributed consensus of second-order multiagent systems with nonconvex input constraints[END_REF][START_REF] Zhou | Constrained consensus in continuoustime multiagent systems under weighted graph[END_REF][START_REF] Dinh | Multiagent system consensus under input and state constraints[END_REF][START_REF] Dinh | Robust consensus analysis and design under relative state constraints or uncertainties[END_REF]. In [START_REF] Nedi | Constrained Consensus and Optimization in Multi-Agent Networks[END_REF], a constrained consensus algorithm and distributed optimization problems were proposed, where agents with state constraints are investigated and they are required to lie in individual closed convex sets. In another work, [START_REF] Wei | Consensus problems for linear time-invariant multi-agent systems with saturation constraints[END_REF] studied a consensus problem of simple integrator MASs under input constraints. Following this research line, a distributed consensus of secondorder MASs with nonconvex input constraints was addressed in [START_REF] Mo | Distributed consensus of second-order multiagent systems with nonconvex input constraints[END_REF]. It is shown that the input constrained consensus achieved if the graph has a directed spanning tree. Another direction to deal with input and state constraints, discarded consensus algorithms are employed [START_REF] Zhou | Constrained consensus in continuoustime multiagent systems under weighted graph[END_REF]. Moreover, in order to achieve the global consensus in the presence of agents' inputs, states, or relative state constraints [START_REF] Dinh | Multiagent system consensus under input and state constraints[END_REF][START_REF] Dinh | Robust consensus analysis and design under relative state constraints or uncertainties[END_REF], the MASs is reformulated in form of a network of Lure systems. However, the formation problem under state constraints in the clustered network is still an open problem.

On the other hand, it has been noticed that interaction among agents in the aforementioned networks is either continuoustime or discrete-time. However, due to either energy constraints occurring in long-time interactions or communication constraints, agents can only impulsively exchange information with their neighbors or be subjected to abrupt changes at specific instants [START_REF] Hong Guan | Consensus analysis based on impulsive systems in multiagent networks[END_REF][START_REF] Zhi-Hong Guan | Impulsive consensus algorithms for second-order multi-agent networks with sampled information[END_REF][START_REF] Hu | Second-order consensus of multi-agent systems in the cooperationcompetition network with switching topologies: A time-delayed impulsive control approach[END_REF][START_REF] Liu | Consensus seeking in multi-agent systems via hybrid protocols with impulse delays[END_REF]. This leads to a hybrid interaction that combines both continuous and discrete interactions among agents. In [START_REF] Hong Guan | Consensus analysis based on impulsive systems in multiagent networks[END_REF], a sufficient result has been derived for the impulsive consensus of first-order MASs, where the graphs of continuous-time and impulsive-time topologies contain a spanning tree. Following this research line, there are several types of research [START_REF] Zhi-Hong Guan | Impulsive consensus algorithms for second-order multi-agent networks with sampled information[END_REF][START_REF] Hu | Second-order consensus of multi-agent systems in the cooperationcompetition network with switching topologies: A time-delayed impulsive control approach[END_REF], which have dealt with the consensus problem of the second-order MASs under an impulsive control strategy. Moreover, inspired by the results in [START_REF] Jadbabaie | Coordination of groups of mobile autonomous agents using nearest neighbor rules[END_REF][START_REF] Ren | Consensus seeking in multiagent systems under dynamically changing interaction topologies[END_REF], the necessary condition of consensus on graph connections among agents may require. This is investigated in [START_REF] Liu | Consensus seeking in multi-agent systems via hybrid protocols with impulse delays[END_REF], where the first-order MASs with hybrid delay consensus protocols are described in the form of impulsive systems. In other directions on consensus problem under hybrid communication, the network is partitioned into several groups or clusters [START_REF] Cesar Bragagnolo | Reset strategy for consensus in networks of clusters[END_REF][START_REF] Irinel-Constantin Morarescu | Coordination in Networks of Linear Impulsive Agents[END_REF]. The works in [START_REF] Cesar Bragagnolo | Reset strategy for consensus in networks of clusters[END_REF] proposed a quasi-periodically reset strategy and provided some LMI conditions to guarantee the globally uniformly exponential consensus for MASs, where the intra-cluster communication structures are represented by directed and strongly connected graphs. The researches in [START_REF] Irinel-Constantin Morarescu | Coordination in Networks of Linear Impulsive Agents[END_REF] investigated the sufficient conditions for event-triggered consensus. Moreover, both of state and output consensus problem in the clustered network of generic linear MASs are respectively investigated in [START_REF] Van Thiem Pham | Consensus of Multi-Agent Systems in Clustered Networks[END_REF][START_REF] Van Thiem Pham | Impulsive Observer-Based Control in Clustered Networks of Linear Multi-Agent Systems[END_REF] and [START_REF] Van Thiem Pham | Adaptive Output Consensus Design in Clustered Networks of Heterogeneous Linear Multi-Agent Systems[END_REF]. However, in most of the above, constraints on the states of agents are not considered.

As a practical example, let us analyze two groups of multi-UAV. To support the cooperative flying, each UAV is equipped with a wireless device to communicate with other UAVs in the same group and on-board sensors to measure or estimate its absolute position and speed. Formation problem naturally appears when two groups are executing the same mission, e.g., scanning an area or protecting civil. Because of sensors' limited ranges and limited speeds of UAVs, the measurements of UAVs' positions and velocities are subject to saturation constraints. Moreover, due to the short-range communications of wireless technologies, it is impossible for each UAV to leave its group and continuously communicate with other UAVs in other groups. Therefore, it is more practical for the collaboration of these groups if each group has one leading UAV to communicate with other groups' leading UAVs through a cellular network at some specific discrete moments. As a result, the information exchange in these networks is not only continuous but also discrete.

Motivated by both theoretical and practical issues men-tioned above, this paper investigates the state formation control problem under state constraints in clustered MASs where agents have generic linear dynamics. Our approach covers broader systems and scenarios than those in the existing studies [START_REF] Cesar Bragagnolo | Reset strategy for consensus in networks of clusters[END_REF][START_REF] Irinel-Constantin Morarescu | Coordination in Networks of Linear Impulsive Agents[END_REF]. Next, a robust formation protocol, which deals with the continuous-time communication inside clusters and discrete-time information exchange between clusters, is introduced. Compared with the previous results [START_REF] Van Thiem Pham | Consensus of Multi-Agent Systems in Clustered Networks[END_REF][START_REF] Van Thiem Pham | Impulsive Observer-Based Control in Clustered Networks of Linear Multi-Agent Systems[END_REF], the protocol is more practical and complicated. It is then shown that the considered robust formation control problem can be indirectly solved by studying the robust stability of an equivalent system by matrix theory and algebraic graph theory. In comparison with the one in [START_REF] Cesar Bragagnolo | Reset strategy for consensus in networks of clusters[END_REF][START_REF] Van Thiem Pham | Consensus of Multi-Agent Systems in Clustered Networks[END_REF][START_REF] Van Thiem Pham | Impulsive Observer-Based Control in Clustered Networks of Linear Multi-Agent Systems[END_REF], our approach shows the important role of communication between leaders at some specific discrete instants, represented by the stochastic matrix. Accordingly, a sufficient condition will be derived in terms of LMIs for the robust distributed formation of clustered networks of generic linear agents under state constraints and hybrid communications.

The rest of the paper is organized as follows. Section II is dedicated to problem formulation. In Section III, the robust formation problem of MASs under state constraints in a hybrid clustered network is investigated. A possible application of the proposed control law to the UAVs formation flying is illustrated in Section IV. Finally, conclusions are provided in Section V.

Notations and symbols. N denotes the sets of non-negative integers. Real, and non-negative real numbers are denoted by R, and R + , respectively. 0 N denotes the zero matrices with appropriate dimensions. 1 N denotes the (N × 1) column vectors whose elements are all ones. A matrix is called SIA if it is stochastic, irreducible, and aperiodic. On the other hand, diag{} represents diagonal or block-diagonal matrices, and sym(A) denotes A + A T for any real matrix A. Lastly, • and ⊗ stands for the Hadamard and Kronecker product.

Problem Formulation

Graph Theory

An undirected graph G consists of a pair (V, E), where the vertex set V = {v 1 , • • • , v N } denotes the set of nodes and the edge set E ⊆ V × V represents the interactions. The adjacency matrix is defined as A = [a (i j) ], with a (i j) > 0 if (v j , v i ) ∈ E, and

a (i j) = 0 otherwise. The Laplacian matrix L = [L (i j) ] ∈ R N×N is defined as L (ii) = j i a (i j) ; L (i j) = -a (i j) .
In the sequel, we consider that the network

G is subdivided into m undirected subnetworks C τ , ∀τ ∈ {1, • • • , m} represented by the graphs G 1 , • • • , G m such that G 1 = (V 1 , E 1 ), • • • , G m = (V m , E m ), where V = V 1 ∪V 2 ∪. . .∪V m and V τ ∩V g = for all τ, g = 1, • • • m, τ g and E = E 1 ∪ E 2 • • • ∪ E m .
The communication graph of each subnetwork G τ is represented by a Laplacian matrix L τ . Each cluster has a specific agent called the leader, and denoted in the following by l τ ∈ V τ , ∀τ ∈ {1, • • • , m}. The remaining agents are called followers and are denoted by f h . The set of leaders will be denoted by

I = {l 1 , • • • , l m }. At par- ticular time instant t k , k ∈ N, t k ≥ 0 of a time sequence {t k } that satisfies t 1 < t 2 < • • • , lim t k →∞ t k = ∞,
the leaders interact following a predefined interaction map E l ⊂ I × I. The leaders communication graph G l = (I, E l ) is also supposed to be a undirected graph. Finally, without loss of generality, each leader is considered as the first agent of its cluster [START_REF] Van Thiem Pham | Adaptive Output Consensus Design in Clustered Networks of Heterogeneous Linear Multi-Agent Systems[END_REF].

C τ = {l τ , f m τ-1 +2 , • • • , f m τ }, ∀τ ∈ {1, • • • , m}, (1)
where m 0 = 0, m m = N and the cardinality of C τ is given by

C τ = n τ = m τ -m τ-1 , ∀τ ≥ 1.

Network Dynamics

We consider a group of N linear identical agents that interact in m clusters. The dynamics of each agent i is described by

ẋi = Ax i + Bu i , (2)
where

x i = [x i,1 , • • • , x i,n ] T ∈ R n is the state, u i ∈ R p is the control input; A ∈ R n×n , B ∈ R n×p and C ∈ R q×n .
According to (1), we concede that each agent has a vector state denoted by

x l τ = [x l τ,1 , • • • , x l τ,n ] T ∈ R n for the lead- ers l τ and x f h = [x f h,1 , • • • , x f h,n ] T ∈ R n for the followers f h , ∀τ h = 1, • • • , N. The desired state formation is denoted by R = [r 1 , r 2 , • • • , r N ] ∈ R Nn , where r i ∈ R n , ∀i ∈ V
is a formation variable of agent i. Therefore, the desired state formation R is meaning that agents in a network need to achieve a reference coordinate frame corresponding to their positions and velocities.

However, in practice, the measurement part may have bounded nonlinearities or saturation constraints due to sensor limitations as well as physical limitations. Moreover, different agents inside each cluster may have different interactive capabilities. Therefore, the upper and lower bounds of saturation constraints may also be different. This leads to the heterogeneous asymmetric saturation levels.

Because only the leaders of these clusters can communicate together at some reset times t k , a formation protocol for a clustered network with state constraints based on local information is given by

u i = K N j=1 a (i j) φ j (z j (t)) -φ i (z i (t)) + Qr i , t ∈ (t k , t k+1 ) (3)
where z i (t) = x i (t)r i denote the state formation variables of agent i, and K, Q ∈ R p×n denote the protocol gain matrices, respectively. The general continous functions

φ i (z i ) : R n → R n , i = 1, • • • , N satisfying the following sector-bounded con- ditions (φ i (z i (t)) -Υ 1 z i (t)) • (φ i (z i (t)) -Υ 2 z i (t)) ≤ 0 ∀z i (t) ∈ R n , (4)
where

Υ 1 = diag{υ k,1 } k=1,••• ,n ∈ R n , Υ 2 = diag{υ k,2 } k=1,••• ,n ∈ R n
are matrices composed of known sector slopes υ k,1 < υ k,2 . An example of φ i is the saturation function s i (•) defined by

s i (z i (t))          s i : z i (t) ≤ s i , z i (t) : s i < z i (t) < s i , s i : z i (t) ≥ s i (5)
where s i ≤ 0 ≤ s i , s i < s i are known constants which are called saturation levels.

The exchanged information between leader's states at the reset time t k can be described by

∆z l i (t k ) = m j=1 a l(i j) (z l j (t k ) -z l i (t k )), t = t k , (6)
where

∆z l i (t k ) = z l i (t + k ) -z l i (t - k )
, where z l i (t + k ) and z l i (t - k ) represent the right and left limit of z l i at t k , respectively. Without loss of generality, we assume that z l i (t - k ) = z l i (t k ). Furthermore, a l(i, j) is the (i, j)th entry of the weighted adjacent matrix A l = [a l(i j) ], and m denotes the set of leaders in graph G l . The Laplacian matrix

L l = [L l(i j) ] ∈ R m×m is defined as L l(ii) = m j i=1 a l(i j) ; L l(i j) = -a l(i j) .
Then the collective dynamics of system (2) under the consensus protocol [START_REF] Guo | Local control strategy for moving-target-enclosing under dynamically changing network topology[END_REF] and the interaction between leaders (6) can be rewritten as

żi (t) = Az i (t) -BK N j=1 L (i j) w j (t) + (A + BQ)r i , t ∈ (t k , t k+1 ), ∆z l i (t k ) = - m j=1 L l(i j) z l j (t k ), t = t k , (7)
w i (t) = φ i (z i (t)).
It can be seen that the evolution of the dynamic system described by ( 7) is influenced by the events that happen at the reset moments t k . Thus, the evolution of the whole system can be viewed as a hybrid system that evolves as follows: during interval time (t k-1 , t k ), the interaction among nodes in each cluster C τ is related only to the graph G τ . Moreover, at each instant time t k , the leaders update their states instantaneously according to the topology of G l , and thus, there exit jump phenomena in the states' leader, while the states' followers in a cluster are recalculated based on the new information's leader. The objective of the above collaboration is to achieve a specific global objective namely state formation defined as follows.

Definition 1. The MASs (2) is said to be achieve state formation anticipated by R if there is vector h(t) ∈ R n such that

lim t→∞ (x i (t) -r i -h(t)) = 0, ( 8
)
for any given bounded initial condition.

For the sake of clarity, an example, as depicted in Fig. 1, is given to illustrate this kind of problem. There are 6 agents divided into two clusters, red and black, that have to realize a hexagon formation. Each agent receives only the state information of its neighbors in the same subnetwork (the same color). If there is no communication between subnetworks (for example, agents 2 and 5 or agents 3 and 6), then the 6-agent network cannot achieve the desired formation. Therefore, in order to ensure the task of 6 agents, at some discrete-time instants, a communication between one red and one black agent (called leader 1 and leader 2, respectively) is activated.

Next, the following assumptions are utilized. A2. a l(i j) > 0, m j i=1 a l(i j) < 1.

A3. The matrix pair (A, B) is stabilizable.

Remark 1. Assumption A1 is needed to guarantee that the Laplacian matrix

L τ of G τ , ∀τ ∈ {1, • • • , m} satisfies the follow- ing proprieties L τ 1 N τ = 0, r T τ L τ = 0 and r T v 1 N τ = 1
, where 1 N τ , and r T τ = 1 N τ 1 N τ are the right and left eigenvectors of L τ associated with zero eigenvalue, respectively. Assumption A2 ensures the matrix P l = I -L l is a stochastic matrix with positive diagonal elements (more information is shown in (9)). Moreover, assumption A3 is for the existence of a controller. Definition 2. [START_REF] Jadbabaie | Coordination of groups of mobile autonomous agents using nearest neighbor rules[END_REF] Let F p a nonnegative and square matrix whose row sums are all equal 1 (i.e., F p 1 = 1). Then, matrix F p is called row stochastic. Hereafter, the time index t is omitted in expressions of x i , h and other variables just for conciseness of mathematical representations. Now, the considered problem in the current research is stated as follows.

Hybrid Robust Formation Control Problem: Consider a group of N agents with their auxiliary systems defined in [START_REF] Oh | Formation Control and Network Localization via Orientation Alignment[END_REF], and suppose that assumptions A1-A3 hold. Design the hybrid robust distributed formation control (3) such that the collective dynamics of the clustered MAS ( 7) satisfies Definition 1.

Robust Distributed Formation Design

In this section, we propose a solution for the above hybrid robust formation control problem with fixed topologies G τ and G l satisfying assumption A1. The proposed design is composed of two steps. First, by employing results from matrix theory and algebraic graph theory, we show that the considered problem can be indirectly solved by the robust stability of an equivalent system. Then, the robust stability design of the equivalent system is derived in terms of LMIs.

Prerequisites

From the Ger šgorin theorem [START_REF] Olfati | Consensus problems in networks of agents with switching topology and time-delays[END_REF], we know that λ m ≤ 2d max (G l ), where λ m is the largest eigenvalue of the Lalapcian of the graph G l , and d max (G l ) is the maximum out-degree of the nodes of G l , where

deg out (v li ) = m i=1 a l(i j)
Therefore, we can get 0 < λ m < 2. Let us introduce P l(i j) = -L l(i j) = a l(i j) > 0,

P l(ii) = 1 -L l(ii) = 1 - m j i=1 a l(i j) > 0, (9)
then m j=1 P l(i j) = 1, and P l = I -L l is a row stochastic matrix with positive diagonal elements, and according to A1 and A2, it has an eigenvalue λ 1 = 1 with algebraic multiplicity equal to one, and all the other eigenvalues satisfy 0

< |λ i | < 1, i = 2, • • • , m.
Moreover, as mentioned above the network is subdivided into m undirected subnetworks. Then, L ∈ R N×N stands for the Laplacian matrix associated with the graph G, which has a diagonal form

L =             L 1 • • • 0 . . . . . . . . . 0 • • • L m             . ( 10
)
Now, some algebraic properties of L are presented in the following Proposition. Proof. See in [START_REF] Van Thiem Pham | Impulsive Observer-Based Control in Clustered Networks of Linear Multi-Agent Systems[END_REF].

Next, it follows (9), the system (7) can be written in a form of the overall network dynamics

ż = (I N ⊗ A)z -(L ⊗ BK)w + (I N ⊗ A)R, t ∈ (t k , t k+1 ) z l (t + k ) = (P l ⊗ I n )z l (t k ), t = t k (11)
w = Φ(z),

where

P l = I -L l , A = A + BQ, Φ(z) = [φ T 1 (z 1 ), • • • , φ T N (z N )] T and z = [z T l 1 , z T f 2 , • • • , z T f m 1 , • • • , z T l m , • • • , z T f mm = z T f N ] T ∈ R Nn , z l = [z T l 1 , • • • , z T l m ] T ∈ R mn .
containing respectively the states of agents and leader's states.

Let us introduce the extended stochastic matrix P e as follows

P e = M T P l 0 0 I N-m M ∈ R N×N , (12)
where M is a permutation matrix.

Then, the second equation in [START_REF] Yang | Fault-tolerant Consensus of Leader-following Multi-agent Systems Based on Distributed Fault Estimation Observer[END_REF] can be expressed by

z(t + k ) = (P e ⊗ I n )z(t k ), t = t k .
In the following, let U ∈ R N×N be an orthogonal matrix, and employing Proposition 1, we obtain

U -1 LU = 0 m 0 0 Γ = Λ ∈ R (N)×(N) , (13)
Γ = diag{γ m+1 , • • • , γ N } ∈ R (N-m)×(N-m) .
Finally, let us also introduce the new variable

ψ = (U -1 ⊗ I n )z. ( 14
)
It follows the variable ψ in ( 14), we now formulate our statement as the following

ψ = (I N ⊗ A)ψ -(ΛU -1 ⊗ BK)w + HR, t ∈ (t k , t k+1 ) ψ(t + k ) = (P ψ ⊗ I n )ψ(t k ), t = t k , (15)
z = (U ⊗ I n )ψ, w = Φ(z),
where P ψ = U -1 P e U and H = (U -1 ⊗ A).

In the next part of this paper, thanks to results from matrix theory and algebraic graph theory, we show that the robust formation control problem of MASs in clustered network [START_REF] Yang | Fault-tolerant Consensus of Leader-following Multi-agent Systems Based on Distributed Fault Estimation Observer[END_REF] is indirectly solved by considering the robust stability of the system (15).

Formation Analysis in Clustered Network

In order to simplify the presentation of the next results let us partition the matrices U -1 , U into

U -1 = [U T 3 U T 4 ] T , U = [U 1 U 2 ], (16)
where

U T 3 ∈ R m×N , U T 4 ∈ R (N-m)×N and U 1 ∈ R N×m , U 2 ∈ R N×(N-m) . U 1 =              1 N 1 • • • 0 . . . . . . . . . 0 • • • 1 Nm              , U T 3 =              r T 1 • • • 0 . . . . . . . . . 0 • • • r T m              , (17)
which satisfies LU 1 = 0 N×n , U T 3 L = 0 m×N . This allows to decompose [START_REF] Nedi | Constrained Consensus and Optimization in Multi-Agent Networks[END_REF] into two parts :

ψ 1 = (U T 3 ⊗ I n )z, ψ 2 = (U T 4 ⊗ I n )z, (18)
where ψ 1 ∈ R mn and ψ 2 ∈ R Nn-mn . Now we are able to introduce the first main results of this paper.

Theorem 1. Consider the overall network dynamics system (15) satisfying A1-A3 the hybrid robust formation control problem is solved if the following formation feasibility condition holds

(A + BQ)(r i -r j ) = 0, ∀i, j = 1, • • • , N. (19)
and

lim t→∞ ψ 2 → 0 (20)
for any given bounded initial conditions.

Proof. If the condition ( 19) holds, then one has that

[L ⊗ (A + BQ)]R = 0. (21)
Pre-multiplying both sides of ( 21) with (U -1 ⊗ I n ) yields

[ΛU -1 ⊗ (A + BQ)]R = 0. ( 22
)
Then pre-multiplying both the sides of ( 22) with

0 m 0 0 Γ -1 ⊗ I n
gives us [U -1 ⊗(A+ BQ)]R = 0, which is equivalent to HR = 0. Therefore, the system (15) leads to the following system

ψ = (I N ⊗ A)ψ -(ΛU -1 ⊗ BK)w, t ∈ (t k , t k+1 ) ψ(t + k ) = (P ψ ⊗ I n )ψ(t k ), t = t k , (23)
z = (U ⊗ I n )ψ, w = Φ(z).
In the following, we firstly show that ψ 1 reaches a constant value, which depends on the dynamics of agents, the graph of each cluster, the interaction between leaders, and the initial conditions. Second, based on the analysis of the first step, the hybrid robust formation control problem, satisfying Definition 1, is solved. First, by employing ( 13), ( 18) and ( 23), the dynamics of ψ 1 can be represented as:

ψ1 = (I m ⊗ A)ψ 1 , (24)
ψ 1 (t + k ) = (U T 3 P e U 1 ⊗ I n )ψ 1 (t k ) + (U T 3 P e U 2 ⊗ I n )ψ 2 (t k ).
Then, the solution of ( 24) with initial condition ψ 1 (t 0 ) = ψ 10 can be obtained by

ψ 1 = e (I m ⊗A)(t-t k ) ψ 1 (t + k ) = (I m ⊗ e A(t-t k ) )ψ 1 (t + k ), (25)
and if lim t→∞ ψ 2 → 0, then ψ 1 (t + k ) can be expressed as:

ψ 1 (t + k ) = lim k→∞ k i=1 (U T 3 P e U 1 ⊗ I n )e (I m ⊗A)(t i -t i-1 ) ψ 1 (t 0 ) = lim k→∞ (U T 3 P e U 1 ) k ⊗ e A(t k -t 0 ) ψ 1 (t 0 ). ( 26
)
In the following, by using results from Lemma 1 and Lemma 2, we prove that matrix U T 3 P e U 1 ∈ R m×m is a row stochastic matrix with positive diagonal elements.

According to (9), the extended stochastic matrix P e in ( 12) can be re-expressed as follows

P e =                        P l 11 0 • • • P l 1m 0 0 I N 1 -1 • • • 0 0 . . . . . . . . . . . . . . . P l m1 0 • • • P lmm 0 0 0 • • • 0 I Nm-1                        ∈ R N×N , (27)
where

P l =             P l 11 • • • P l 1m . . . . . . . . . P l m1 • • • P l mm             =             P l 1 . . . P l m             ∈ R m×m , (28)
and the matrix U 1 ∈ R N×m and U T 3 ∈ R m×N are given in [START_REF] Zhou | Constrained consensus in continuoustime multiagent systems under weighted graph[END_REF]. Now, the matrix P e U 1 is calculated as

P e U 1 =                        P l 11 • • • P l 1m 1 N 1 -1 0 (N 1 -1)×(N 1 -1) . . . . . . P l m1 • • • P lmm 1 Nm-1 0 (Nm-1)×(Nm-1)                        =              E 1 . . . E m              , (29)
where ∀τ ∈ {1, • • • , m}, and

E τ = P l τ1 • • • P l τm 1 N τ -1 0 (N τ -1)×(N τ -1) ∈ R N τ ×m . (30)
Then, the matrix U T 3 P e U 1 is determined as follows

U T 3 P e U 1 =              r T 1 • • • 0 . . . . . . . . . 0 • • • r T m                           E 1 . . . E m              =              r T 1 E 1 . . . r T m E m              , ( 31
)
where

r T τ = [r τ 1 , • • • , r τ Nτ ] ∈ R 1×N τ , ∀τ ∈ {1, • • • , m}, and
r T τ E τ = [r τ 1 P l τ1 + r τ 2 + • • • + r τ Nτ , r τ 1 P l τ2 , • • • , r τ 1 P l τm ].
The sum of the row matrix r T τ E τ is calculated by

N τ k=1 r T τ E τ = r τ 1 P l τ1 + N τ k=2 r τ k + r τ 1 N τ k=2 P l τk . (32)
According to A1, A2 and ( 9), P l τ1 = 1 -m k=2 P l τk , then

N τ k=1 r T τ E τ = N τ k=1 r τ k = 1, (33)
and P l τk > 0, then

r τ 1 P l τ1 + r τ 2 + • • • + r τ Nτ > 0, r τ 1 P l τ2 > 0, • • • , (34)
r τ 1 P l τm > 0.
Subsequently, by employing ( 33) and (34), and according to Definition 2, we see that the matrix U T

3 P e U 1 ∈ R m×m is a row stochastic matrix with positive diagonal elements.

Furthermore, by employing ( 9), and ∀i, τ ∈ {1, • • • , m} Eq. (34) becomes

1 + r τ 1 (P l τ1 -1) = 1 -r τ 1 m i j=1 a l(i j) , r τ 1 P l τ2 = r τ 1 a l i2 , (35) • • • r τ 1 P l τm = r τ 1 a l im .
then the (i, j) th entry of U T

3 P e U 1 is r τ 1 a l(i j) , which implies that the graph G l and the graph of U T

3 P e U 1 have the same edge set. Thus, the graph of the matrix U T

3 P e U 1 is undirected and connected. It means that the graphs of U T

3 P e U 1 has at least one spanning tree.

Based on the above analysis, we showed that the matrix U T

3 P e U 1 is a row stochastic matrix with positive diagonal elements and its graph has at least one spanning tree. Then, according to Lemma 2, the matrix U T

3 P e U 1 is SIA. Therefore, from Lemma 1, there exits a column vector c T such that lim k→∞

(U T 3 P e U 1 ) k = 1 m c T . ( 36
)
Then, by substituing (36) and ( 26) into ( 25), one has

ψ 1 = 1 m c T ⊗ e A(t-t 0 ) ψ 10 . ( 37
)
Second, by introducing the variables

µ 1 = (U ⊗ I n ) ψ 1 0 , µ 2 = (U ⊗ I n ) 0 ψ 2 , ( 38
)
one has z = µ 1 + µ 2 . Then, according to (38) the variable µ 1 is written such as

µ 1 = [U 1 ⊗ I n U 2 ⊗ I n ] ψ 1 0 = (U 1 ⊗ I n )ψ 1 . ( 39
)
It follows that µ 2 = z -µ 1 , where µ 2 = (U 2 ⊗ I n )ψ 2 . And, by using (37) and (39), we obtain

µ 1 = U 1 1 m c T ⊗ e A(t-t 0 ) ψ 10 , = 1 N c T ⊗ e A(t-t 0 ) ψ 10 . (40)
If it follows lim t→∞ ψ 2 → 0 and recalling that (U ⊗ I n ) is nonsingular, then follows from (38) that lim t→∞ µ 2 → 0. Finally, it follows from (40) that

lim t→∞ µ 2 = lim t→∞ (z -µ 1 ) = lim t→∞ (x -R -1 N c T ⊗ e A(t-t 0 ) ψ 10 h(t) ) → 0, = lim t→∞ (x i -r i -c T ⊗ e A(t-t 0 ) ψ 10 h(t) ) → 0, ( 41
)
which implies that the system (11) can achieve state formation anticipated by R, meaning the hybrid robust formation control problem was solved. This completes the proof.

Remark 2. h(t) in [START_REF] Dong | Timevarying formation control for unmanned aerial vehicles: Theories and applications[END_REF] generally can be used to guide a group of agents to achieve an anticipated formation specified by R as shown in Fig. 2 and it is considered as the formation position function. Moreover, the formation function h(t) in considered clustered network is also described as (41), which depend on agents initial states and formation vector, agent's dynamics, communication networks' cluster and leaders.

Remark 3. According to Thorem 1, one sees that to ensure the state formation R, not only the communication topology is required to be connected and the Laplacian matrix is a symmetric matrix, but also the formation vector should satisfy the constraint [START_REF] Dinh | Robust consensus analysis and design under relative state constraints or uncertainties[END_REF] . Therefore, Theorem 1 establishes the relationship between the formability and the communication topology, the agents' dynamics and the formation vector.

Robust Stabilization Controller Design

Based on the above analysis in Subsection B, the objective now is to design the matrix K ∈ R p×n , such that the system (15) is robustly stable, i.e., lim t→∞ ψ 2 → 0. The design of such robust stabilization controller gain K is given in the following theorem.

Theorem 2. Consider the system (15) satisfying assumptions A1-A3 and condition [START_REF] Dinh | Robust consensus analysis and design under relative state constraints or uncertainties[END_REF]. It is robustly stable if there exist positive-definite and diagonal matrices P, Π, Z ∈ R n×n and X ∈ R p×n such that the following LMIs are feasible,

Ξ 1 γ 2 BX + (Υ 1 -Υ 2 ) 2 Z * -Z ≤ 0, (42)
Ξ 2 -γ N BX + (Υ 1 -Υ 2 ) 2 Z * -Z ≤ 0, (43)
Z P P Π -1 ≥ 0, (44)
where

Ξ 1 = sym(AP + γ 2 BXΥ 2 ) + αP, γ 2 = min{Γ}, Ξ 2 = sym(AP + γ N BXΥ 2 ) + αP, γ N = max{Γ}. Furthermore, K = XP -1 .
Proof. Firstly, we define

V = V(ψ) = ψ T (Θ ⊗ P -1 )ψ, (45)
where

P = P T > 0, Θ = 0 m 0 0 I N-m .
Obviously, V is positive semi-definite. If we can prove that between impulses t k and t k+1 , the function V is decreasing

V < 0, ∀t ∈ (t k , t k+1 ), (46)
where

V =ψ T [Θ ⊗ (A T P -1 + P -1 A)]ψ- -ψ T [(ΘΛU -1 + UΛΘ) ⊗ P -1 BK]w. (47)
and at reset time

t k V(t k ) ≥ V(t + k ). (48)
Then according to the Lasalle's invariance principle, ψ(t) globally exponentially converges to the largest invariance set contained in {ψ ∈ R Nn | V(ψ) = 0} for any initial conditions. It can be seen from ( 47) and definition of matrix Θ in (45) that V(ψ) = 0 if and only if lim t→∞ ψ 2 → 0, where

ψ 2 = [ψ m+1 , • • • , ψ N ] T ∈ R Nn-mn .
In the following, the condition (46) is equivalent to V + αV ≤ 0, where α > 0. Thus, Eq. (47) becomes

V + αV = ψ T [Θ ⊗ (A T P -1 + P -1 A + αP -1 )]ψ- -ψ T [(ΘΛU -1 + UΛΘ) ⊗ P -1 BK]w ≤ 0. (49)
Then, using the S-procedure and the sector-bounded conditions (4), there exits a diagonal matrix Π ∈ R n×n , Π ≥ 0, such that

V + αV - N j=1 (w j -Υ 1 z j ) T Π(w j -Υ 2 z j ) ≤ 0. ⇔ V + αV -w T (I N ⊗ Π)w -z T (I N ⊗ ΠΥ 1 Υ 2 )z+ + w T [I N ⊗ Π(Υ 1 + Υ 2 )]z ≤ 0. (50) Then, with z = (U ⊗ I n )ψ, U T = U -1 E.q (50) becomes V + αV -w T (I N ⊗ Π)w -ψ T (I N ⊗ ΠΥ 1 Υ 2 )ψ+ + w T [U ⊗ Π(Υ 1 + Υ 2 )]ψ ≤ 0. ⇔ ψ w T Ψ 1 Ψ 2 Ψ 3 Ψ 4 ψ w ≤ 0. ⇔ Ψ 1 Ψ 2 Ψ 3 Ψ 4 ≤ 0. ( 51
)
where

Ψ 1 = Θ ⊗ (A T P -1 + P -1 A + αP -1 ) -I N ⊗ ΠΥ 1 Υ 2 , Ψ 2 = -ΘΛU -1 ⊗ P -1 BK + 1 2 U T ⊗ Π(Υ 1 + Υ 2 ), Ψ 3 = -UΛΘ ⊗ K T B T P -1 + 1 2 U ⊗ Π(Υ 1 + Υ 2 ), Ψ 4 = -I N ⊗ Π.
Subsequently, taking the Schur complement to (51) results in

Ψ 1 -Ψ 2 Ψ -1 4 Ψ 3 ≤ 0, in which -Ψ 2 Ψ -1 4 Ψ 3 = ΘΛ 2 Θ ⊗ P -1 BKΠ -1 K T B T P -1 + + 1 4 I N ⊗ Π(Υ 1 + Υ 2 ) 2 + 1 2 ΛΘ ⊗ (Υ 1 + Υ 2 )K T B T P -1 + + 1 2 ΘΛ ⊗ P -1 BK(Υ 1 + Υ 2 ).
Next, by considering Ψ 1 -Ψ 2 Ψ -1 4 Ψ 3 ≤ 0, and since Λ, and Γ in ( 13) are diagonal, one obtains

(A T P -1 + P -1 A + αP -1 ) -ΠΥ 1 Υ 2 + + γ 2 k P -1 BKΠ -1 K T B T P -1 + 1 4 Π(Υ 1 + Υ 2 ) 2 + (52) + 1 2 γ k (Υ 1 + Υ 2 )K T B T P -1 + 1 2 γ k P -1 BK(Υ 1 + Υ 2 ) ≤ 0, where γ k , k = m + 1, • • • , N.
After, multiplying both sides (52) with P, we get

PA T + AP + αP + 1 4 PΠ(Υ 1 -Υ 2 ) 2 P+ + γ 2 k BKΠ -1 K T B T + 1 2 γ k P(Υ 1 + Υ 2 )K T B T + (53) + 1 2 γ k BK(Υ 1 + Υ 2 )P ≤ 0.
Since P, Υ 1 , Υ 2 are diagonal matrices, PΥ 1 = Υ 1 P, PΥ 2 = Υ 2 P, and (53) is equivalent to

PA T + AP + αP + γ k Υ 2 PK T B T + γ k BKPΥ 2 + + 1 4 PΠ(Υ 1 -Υ 2 ) 2 P + γ 2 k BKΠ -1 K T B T + (54) + 1 2 γ k P(Υ 1 -Υ 2 )K T B T + 1 2 γ k BK(Υ 1 -Υ 2 )P ≤ 0.
It leads to

sym(AP + γ k BKPΥ 2 ) + αP+ + [γ k BKP + 1 2 (Υ 1 -Υ 2 )ΠP 2 ]× (55) × (ΠP 2 ) -1 [γ k BKP + 1 2 (Υ 1 -Υ 2 )ΠP 2 ] T ≤ 0.
Taking Z ΠP 2 , Z 0 and K = XP -1 . Then, using the Schur complement again with (55) leads to

Ξ γ k BX + (Υ 1 -Υ 2 ) 2 Z * -Z ≤ 0, (56)
where Ξ = sym(AP + γ k BXΥ 2 ) + αP and γ k , k = m + 1, • • • , N are eigenvalues of Laplacian matrix L.

Since γ 2 = min{Γ}, γ N = max{Γ} and γ 2 ≤ γ m+1 ≤ • • • ≤ λ N , we can represent γ p , p = m+2, • • • , N-1 as convex combination of γ 2 and γ N . Thus, we derive (42) and (43). The LMI (44) is obtained straigthforward from Z ΠP 2 , Z 0.

On the other hand, at the reset time t = t k one has

V(ψ(t + k )) -V(ψ(t k )) = ψ(t k ) T [(P ψ ⊗ I n ) T (Θ ⊗ P -1 (P ψ ⊗ I n )- -(Θ ⊗ P -1 )]ψ(t k ).
Then, to guarantee the second condition (48), one needs

(P ψ ⊗ I n ) T (Θ ⊗ P -1 )(P ψ ⊗ I n ) -(Θ ⊗ P -1 ) ≤ 0. ( 57
)
by multiplying both sides Eq. ( 57) by (I N ⊗ P) > 0, we obtain

(P ψ ⊗ I n ) T (Θ ⊗ P)(P ψ ⊗ I n ) -(Θ ⊗ P) ≤ 0. ( 58
)
by employing P ψ = U -1 P e U, U -1 = U T and using [START_REF] Li | Robust Distributed Consensus Control of Uncertain Multi-Agents Interacted by Eigenvalue-Bounded Topologies[END_REF], and ( 9), it is easy to verify that P T e ΘP e -Θ ≤ 0. Thus, the condition (48) is always true. Remark 4. In case of homogeneous constraints, the upper and lower sectors and bounds for state constraints of all agents are the same, then Υ 1 and Υ 2 are multiple of identity matrices, i.e., Υ 1 = υ 1 I n and Υ 2 = υ 2 I n . Then, the variable P ∈ R n×n in Theorem 2 is not required to be diagonal. Thus, the associated LMI problem is less conservative and its feasibility would be improved.

Remark 5. According to the LMIs (42)-(44), one sees that the dimension of variables P ∈ R n×n , X ∈ R p×n are just equal to that of the matrix A ∈ R n×n of each agent. Thus, the complexity of those LMI problems is low. If γ 2 , γ N are computed by a given Laplacian matrix L with respect to graph G, then we can solve LMIs (42)-(44) in fully distributed fashion i.e., each agent can compute the gain matrix K by itself and implement the consensus protocol (3) using only local information (its information and its neighbors' information).

Application to Formation of UAVs

In this section, we consisder a group of N UAV's motion in d-dimensional Euclidean space, which is modeled as the second order dynamics in [START_REF] Lafferriere | Decentralized control of vehicle formations[END_REF], where the state variable consists of the configuration states (position-p x i , p y i ) and their derivatives (velocity-v x i , v y i ), the control input u i ∈ R p denotes the acceleration commands. Finally, the system matrices are given such as

A = I d ⊗ 0 1 a 1 21 a 1 22 , B = I d ⊗ 0 1 . (59)
In order to illustrate that the proposed approaches are implemented in the complex network, we consider the following The dynamics of the agents and the Laplacian matrix of leader network G l are given by

A = I 2 ⊗ 0 1 0 -1 , B = I 2 ⊗ 0 1 . L l =         0.9 -0.4 -0.5 -0.4 0.7 -0.3 -0.5 -0.3 0.8         ⇒ P l =         0.1 0.4 0.5 0.4 0.3 0.3 0.5 0.3 0.2         .
In this example, we suppose that agents 3, 6 and 10 have more communication capability than other agents. Thus, we choose them as leaders of clusters 1, 2 and 3, respectively.

The formation considered this time is pentacle. Then, the formation specified by R(m) are given by

r 1 = [0 4 √ 3], r 2 = [2 2 √ 3], r 3 = [4 √ 3 2 √ 3], r 4 = [2 √ 3 0], r 5 = [4 -4 √ 3], r 6 = [0 -2 √ 3], r 7 = [-4 -4 √ 3], r 8 = [-2 √ 3 0], r 9 = [-4 √ 3 2 √ 3], r 10 = [-2 2 √ 3]. and R = [r 1 r 2 r 3 r 4 r 5 r 6 r 7 r 8 r 9 r 10 ] T ⊗ [1 0] T . It is clear from R that r 1 0 , • • • , r 10 0 T = r x 1 r y 1 0 0 , • • • , r x 10 r y 10 0 0 T .
It means that two scenarios related to UAV's positions and velocities are taken into account: ten UAVs will be controlled to reach a regular pentacle formation in the 2D plane corresponding to their positions, and all of ten UAV's velocites will be achieve a common value such as

lim t→∞ (p j -p i ) -(r j -r i ) = 0, lim t→∞ (v j -v i ) = 0.
Each cluster is color coded, where the first cluster is red, second is blue and third is black.

Heterogeneous constraints

Due to a limited range of sensor, and the limited velocity of each UAV, the states of connected agents are bounded i.e., the

u i =K N j=1 a (i j) φ j (z p x j ) -φ i (z p x i ) φ j (z p y j ) -φ i (z p y i ) φ j (v x j ) -φ i (v x i ) φ j (v y j ) -φ i (v y i ) + Q r x i r y i 0 0 , (60)
where z p x i = p x ir x i ; z p y i = p y ir y i . Then, employing (60), the dynamics of agent i can be reformulated as

ṗx i ṗy i vx i vy i = v x i v y i -v x i -v y i + Q r x i r y i 0 0 + + BK N j=1 a (i j) φ j (z p x j ) -φ i (z p x i ) φ j (z p y j ) -φ i (z p y i ) φ j (v x j ) -φ i (v x i ) φ j (v y j ) -φ i (v y i )
.

At t = t k , the interaction of leaders can expressed as

x l (t + k ) = (P l ⊗ I 2 )
x l (t k ) + r l -(P l ⊗ I 2 )r l , where l = 3, 6, 10, and The interaction among leaders occurs at some instant times. These are defined based on some events or particularly demand of systems, which is decided by an operator. In our simulation, we assume that the reset time of the leader's communication is periodic and it happens at each second as depicted in Fig. 4 (upper). The evolution of leaders' states is also depicted in Fig. 4. It is clear that leaders' values are updated at the reset time t k through the communication graph G l .

x l i = p x l i p y l i v x l i v y l i , x l =           x l 1 x l 2 x l 3           , x l 1 = x 3 , x l 2 = x 6 , x l 3 = x 10 . Choosing Q = [ 0
Convergence of both the variable ψ 2 in ( 18) and the Lyapunov function (45) is depicted in Fig. 5. It again verifies that the clustered network achieves formation under robust formation control protocol (3) if lim t→∞ ψ 2 → 0, shown in Theorem 1, and the Lyapunov function satisfies the condtions (46) and (48). Moreover, according to the simulation results depicted respectively in Fig. 6, and Fig. 7, we see that all positions and velocities of UAVs in clustered network reach the pentacle formation and consensus under the state constraints.

Homogeneous constraints

Finally, in oder to investigate the influence of state constraints on the formation performance, we have carried out two simulations. In the first simulation, we consider constraints on positions that belong to [- 15 15](m) and we suppose that the speed shoud be in the interval [-15 8](m/s). The second simuation suppose that the constraints on positions belong to [-3 3](m) and that speed should be in [-3 3](m/s). The obtained simulation resluts are depicted in Fig. 8 and Fig. 9. In both of cases, one can see that agents achieve and keep the desired formation and that the values of state variables are in the defined region. Moreover, one can remark that when the constraints on the speed are stricter (the second case) the achivement of formation takes more time.

Conclusion

In this paper, a novel approach has been proposed to design distributed robust formation controllers for general linear MASs under state constraints with the following features. First, the considered networks are partitioned into clusters, where the communication between agents inside each cluster is continuous, but the cluster leaders interact at some reset times. Second, it is shown that the robust formation design with state constraints can be indirectly solved by considering the stability of an equivalent system. Third, sufficient conditions for the robust stability of this equivalent system were derived from solutions of local convex LMI problems, which can be solved in a distributed manner. A possible application of our proposed approaches to the UAVs formation flying was illustrated.

Our future work will address the case of output constraints in clustered network with switching topology.

Figure 1 :Figure 2 :

 12 Figure 1: The scheme of formation problem

Lemma 1 .

 1 [START_REF] Ren | Consensus seeking in multiagent systems under dynamically changing interaction topologies[END_REF] Let Γ be a compact set consisting of n × n SIA matrices with the property that for any nonnegative integer k and any B 1 ,• • • , B k ∈ Γ (repetitions permitted), k i=1 B i is SIA. Then,given any infinite sequence B 1 , B 2 , • • • (repetitions permitted), of matrices Γ, there exits a column vector c T such that lim k→∞ k i=1 B i = 1c T . Lemma 2. [25] If B = [b i j ] n×n is a stochastic matrix with positive diagonal elements, and the graph associated with B has a spanning tree, then B is SIA.

Proposition 1 .

 1 Let us consider a network of m clusters satisfying assumption A1, with the Laplacian L ∈ R N×N , then rank(L) = Nm and L has m eigenvalues at zero and all the other Nm eigenvalues of the Laplacian L ∈ R N×N are positive.

Figure 3 :

 3 Figure 3: The communication of the network

1 ]

 1 to satisfy Theorem 1, and c = 0.1, Υ 1 = diag{0, 0.1}, Υ 2 = diag{0.1, 0.2}. Then, solving LMIs in Theorem 2, one has the feedback matrix K = [-5.6625 -6.0109].

Figure 4 :Figure 5 :

 45 Figure 4: A reset signal and leaders' states of UAVs

Figure 6 :

 6 Figure 6: Ten-UAVs'positions (x i , y j ) constraints (left) and ten-UAVs'velocities (v x i , v y i ) constraints (right).

Figure 7 :

 7 Figure 7: Pentacle formation of ten UAVs'positions (x i , y i ) (lower) and consensus of ten UAVs'velocities (v x i , v y i ) under state constraints (upper).

Figure 8 :

 8 Figure 8: Ten-UAVs' positions (x i , y j ) under state constraints belonging to [-15 15](m) (upper); Ten-UAVs' positions (x i , y j ) under state constraints belonging to [-3 3](m) (lower).

Figure 9 :

 9 Figure 9: Ten-UAVs' velocities (v x i , v y i ) under state constraints belonging to [-15 8](m/s) (upper); Ten-UAVs' velocities (v x i , v y i ) under state constraints belonging to [-3 3](m/s) (lower).

Table 1 :

 1 The sate constraints of ten agents 1st state s j (m) s j 2nd state state constraints are the saturation function (4) with heterogeneous constraints such as Table1, and the initial conditions of three clusters are randomized.

	s j	s j (m/s)

In t ∈ (t k , t k+1 ), the control protocol now is

Acknowledgment

The authors would like to thank the Editor and all the reviewers for their constructive suggestions.


MBBS Gemma A Figtree
email: gemma.figtree@sydney.edu.au.

Barbara Casadei

Robert Califf

Filippo Crea

Grant Drummond

Jane E Freedman

Tomasz Guzik

David Harrison

Derek J Hausenloy

Joseph A Hill

James Januzzi

Bronwyn A Kingwell

Carolyn S P Lam

Calum A Macrae

Frank Misselwitz

Tetsuji Miura

Rebecca Ritchie

Maciej Tomaszewski

Joseph C Wu

Junjie Xiao

Faiez Zannad

Dphil

A CALL TO ACTION FOR NEW GLOBAL APPROACHES TO CARDIOVASCULAR DISEASE DRUG SOLUTIONS

Keywords: drug discovery, pharmacotherapy, cardiovascular, organoids, precision medicine therapeutic target, drug discovery, precision medicine, multi-omics, atherosclerosis, heart failure

Whilst we continue to wrestle with the immense challenge of implementing equitable access to established evidence-based treatments, substantial gaps remain in our pharmacotherapy armament for common forms of cardiovascular disease including coronary and peripheral arterial disease, heart failure, hypertension and arrhythmia. We need to continue to invest in the development of new approaches for the discovery, rigorous assessment and implementation of new therapies. Currently, the time and cost to progress from lead compound/product identification to the clinic, and the success rate in getting there reduces the incentive for industry to invest, despite the enormous burden of disease and potential size of market. There are tremendous opportunities with improved phenotyping of patients currently batched together in syndromic "buckets". Use of advanced imaging and molecular markers may allow stratification of patients in a manner more aligned to biological mechanisms that can, in turn be targeted by specific approaches developed using high throughput molecular technologies. Unbiased "omic" approaches enhance the possibility of discovering completely new mechanisms in such groups. Furthermore, advances in drug discovery platforms, and models to study efficacy and toxicity more relevant to the human disease are valuable. Reimagining the relationships among discovery, translation, evaluation and implementation will help reverse the trend away from investment in the cardiovascular space, establishing innovative platforms and approaches across the full spectrum of therapeutic development.

Introduction

Cardiovascular disease is the leading cause of death globally, with a resurgence in high income countries, and ongoing rapid rise in low and middle income countries [START_REF] Timmis | European Society of Cardiology: Cardiovascular Disease Statistics[END_REF][START_REF] Virani Salim | Heart Disease and Stroke Statistics-2020 Update: A Report From the American Heart Association[END_REF][START_REF] Roth | Global Burden of Cardiovascular Diseases and Risk Factors, 1990-2019: Update From the GBD 2019 Study[END_REF] . Initiatives to improve equitable access to and implementation of established effective therapies (e.g. statins) continue to be an international challenge. However, substantial gaps remain in our pharmacotherapy armament for common forms of cardiovascular disease, including coronary and peripheral arterial disease, heart failure, hypertension and arrhythmia. Collaborative efforts to tackle access and affordability issues need to be combined with initiatives to accelerate new therapeutic discovery and development. Integration of health economic analyses at all stages of the development and trial pipeline, that take into consideration the perspective of those who pay for healthcare relevant to the different jurisdictions, is a critical factor for successful and equitable implementation. Currently the time and cost to progress from lead compound/product identification to the clinic, and the success rate in getting there reduces the incentive for industry to invest, despite the substantial burden of disease and potential market size. Re-imagining the relationship between discovery, translation and implementation is required to rapidly reverse these trends. This can be achieved by establishing innovative platforms and approaches across the full spectrum of therapeutic development, and re-establishing a thriving and impactful sector capable of discovering and translating new solutions.

Where are we now?

Whilst there have previously been limited specific data on development of cardiovascular drugs, and factors associated with success of new therapies in clinical trials, Hwang and colleagues have provided an excellent quantitative analysis of the field [START_REF] Hwang | Temporal Trends and Factors Associated With Cardiovascular Drug Development, 1990 to 2012[END_REF] . Prior to this, reports had anecdotally suggested that the number of new cardiovascular drugs approved by the US Food and Drug Administration (FDA) had recently declined [START_REF] Fordyce | Cardiovascular drug development: is it dead or just hibernating?[END_REF][START_REF] Darrow | Drug development and FDA approval[END_REF] , and that there was a contraction in the pool of drugs under development targeting cardiovascular disease [START_REF] Pammolli | The productivity crisis in pharmaceutical R&D[END_REF] . Using the large commercial database of drug development activity (Citeline Pharmaprojects, Informa plc), Hwang et al were able to quantify the rate and ultimate success of translation along the pipeline of over 50,000 products for all diseases, and to compare the rates among disease states, as well as examine factors associated with success. These data confirmed that the number of new cardiovascular drugs entering clinical trials at all stages of development declined between 1990 and 2012 (compared with cancer which increased). Many cardiovascular drugs in development were targeted against modifiable risk factors and previously proven mechanisms addressing these risks including antihypertensive agents, lipid modifying agents, and anticoagulants. However, 50% of cardiovascular drugs entering phase III trials between 1990 and 2012 were categorised as targeting novel biological pathways, with this rate of novelty increasing over the time period. Most clinical trials for cardiovascular drugs were sponsored by large pharmaceutical companies. Smaller companies were observed to be active, but more focussed on rarer forms of cardiovascular disease. In all three phases of study, the proportion of cardiovascular trials sponsored by large pharmaceutical companies compared to small-or medium-sized companies was significantly greater than that for non-cardiovascular trials [START_REF] Hwang | Temporal Trends and Factors Associated With Cardiovascular Drug Development, 1990 to 2012[END_REF] .

At the center of the disparity between cardiology and oncology is the lack of routine clinical access to tissue in cardiac and vascular diseases which has led to disease classifications and thus clinical trials focused to a large extent on phenotypes that aggregate relevant visible properties. These phenotypes, while often linked by final common pathways, typically reflect heterogeneous underlying etiologies and thus responses to intervention. Therefore, efforts in cardiology (and other specialties) to reclassify or stratify diseases based on discrete pathogenetic mechanisms have been proposed as one rational approach to resolving the trend away from cardiovascular therapeutic development. Identification of etiologically homogeneous subgroups of patients (endotypes) within a heterogeneous population is a goal of stratified medicine but remains a challenging. For instance, Type 1 myocardial infarction is a phenotype which is caused by different endotypes needing different forms of therapy 8 , which may not be readily defined by current techniques.

Similarly, angina is a phenotype which is caused by different endotypes [START_REF] Kaski | Reappraisal of Ischemic Heart Disease[END_REF] , but we have tended to focus only on the endotypes for which we have efficient interventions (i.e. macrovascular obstruction). This, and some of the additional challenges and potential solutions are outlined below and summarised in Table 1.

Table 1. Challenges and potential solutions for accelerating drug discovery and translation in the cardiovascular space.

Challenges

Potential solutions Historic tendency to target biological mechanisms which have empirically been shown to influence overall outcomes, rather than definitive disease mechanisms (necessary and sufficient) identified in a discrete subset. Thus ~50% agents entering phase III clinical trials still target BP, lipids and coagulation [START_REF] Hwang | Temporal Trends and Factors Associated With Cardiovascular Drug Development, 1990 to 2012[END_REF] , with limited novel therapeutic targets beyond these.

Identification of definitively causal (not association) novel targets and pathways from robust studies of increasingly homogeneous phenotypic subsets through improved use of molecular and advanced imaging and clinical markers. Increasing proportion of CVD drugs entering phase III trials targeting novel causal biologic pathways [START_REF] Hwang | Temporal Trends and Factors Associated With Cardiovascular Drug Development, 1990 to 2012[END_REF][START_REF] Fiuzat | Resourcing Drug Development Commensurate With its Public Health Importance: The Road Ahead[END_REF] Limitations in methods that target known signalling pathways involved in disease processes Advances in drug design platforms will accelerate translation of complex mechanistic knowledge to new therapies. Poor translation of safety and efficacy parameters from preclinical models (especially rodents) to humans Development of mechanistically faithful human exvivo cardiac and vascular organoids, and the creation of similarly robust animal models that do not just mimic the human condition, but rather reflect the definitive mechanisms, and the burden of comorbidities and age. Lack of rigor in some preclinical research (e.g. single centre, not blinding or randomising, no protocol publication, over emphasis of findings) contributes to well-recognised limited reproducibility

Residual risk and "missing" biology

Many medications in cardiovascular disease have been identified through serendipity, for example, none of the major classes of drug treatment for heart failure (renin-angiotensinaldosterone inhibitors, beta blockers, SGLT2 inhibitors), were initially developed for the syndrome. Not surprisingly, drugs discovered in this manner do not appear to address all of the pathophysiologic components of the diseases which they are now used to treat. For example, up to 27% of ST-elevation MI patients do not exhibit traditional modifiable risk factors [START_REF] Vernon | Increasing proportion of ST elevation myocardial infarction patients with coronary atherosclerosis poorly explained by standard modifiable risk factors[END_REF][START_REF] Vernon | ST-Segment-Elevation Myocardial Infarction (STEMI) Patients Without Standard Modifiable Cardiovascular Risk Factors-How Common Are They, and What Are Their Outcomes[END_REF] , and a substantial proportion of patients with coronary artery disease (CAD) progress rapidly to recurrent acute events despite optimal medical management [START_REF] Schiele | Coronary artery disease: Risk stratification and patient selection for more aggressive secondary prevention[END_REF] . The recent successful trial addressing the residual risk driven by inflammation (CANTOS [START_REF] Ridker | Antiinflammatory Therapy with Canakinumab for Atherosclerotic Disease[END_REF] and LoDoCo2 [START_REF] Nidorf | Colchicine in Patients with Chronic Coronary Disease[END_REF] ) are examples of how cardiovascular risk can be further reduced addressing non traditional risk factors. A major limitation of these trials, however, is that they have included the broad population of patients with cardiovascular disease rather than the "endotype"

susceptible to an anti-inflammatory treatment [START_REF] Crea | Addressing Acute Coronary Syndromes: New Challenges and Opportunities After the CANTOS Trial (Canakinumab Anti-inflammatory Thrombosis Outcomes Study)[END_REF] .

There is significant potential to address these gaps through systematic reappraisal of the mechanisms of the constituent disorders, and the application of proven methods of drug discovery to the resultant novel mechanistically robust targets. For quite some time cardiovascular discovery efforts have been dominated by a multitude of signalling pathways, emerging from large-scale profiling technologies, which while they often provide molecular phenotyping [START_REF] Vernon | Utilizing stateof-the-art "omics" technology and bioinformatics to identify new biological mechanisms and biomarkers for coronary artery disease[END_REF] , often lie downstream of the actual causal factors and in many instances may represent compensatory pathways, at least partially explaining the mixed results that have been observed. For example, of the >150 genetic loci reproducibly associated with CAD to date, very few are thought to act via traditional risk factors 19 . Pipelines focused on discoveries from genome-wide association studies (GWAS) have been employed by a number of the leading Pharma companies for over a decade, but new causal pathways and successful interventions based on such targets have been slow to emerge.

Technology to target novel biology

The technologies to efficiently develop new therapeutics once definitive targets have been identified are maturing and approaches including in silico screening, fragment-based drug discovery and tagged libraries have changed the pace of modern drug discovery [START_REF] Murray | The rise of fragment-based drug discovery[END_REF] . An expanded tool kit of molecular targeting through cell, gene, RNA, peptide, and protein (including antibody) therapies and nanoparticle technology provide further diversity of approaches and enhance the likelihood of success against any target [START_REF] Bubb | New opportunities for targeting redox dysregulation in cardiovascular disease[END_REF][START_REF] Landmesser | From traditional pharmacological towards nucleic acid-based therapies for cardiovascular diseases[END_REF] . RNA and gene therapies offer remarkable specificity and safety profiles with successful 'platform' approaches to therapeutic development, for example, the siRNA Inclisiran (Novartis) represents a powerful approach to sustained inhibition of PCSK9 to reduce LDL-cholesterol.

The top 10 (2019) highest growing pharmaceutical companies that are using artificial intelligence (AI) or machine learning for drug discovery have been recently reviewed 23 .

Patients with recurrent events despite best evidence-based care-an opportunity for novel agents

The current unmet need in cardiovascular disease is exemplified by the substantial number of individuals who develop disease or progress despite best practice treatment. Examples include resistant hypertension (estimated to be ~10% of the hypertensive population) and those with advanced atherosclerosis and acute MI, but no elevation in Standard Modifiable cardiovascular Risk Factors ("SMuRF"-less; between 15-30% of ST elevation MI [START_REF] Vernon | Increasing proportion of ST elevation myocardial infarction patients with coronary atherosclerosis poorly explained by standard modifiable risk factors[END_REF][START_REF] Vernon | ST-Segment-Elevation Myocardial Infarction (STEMI) Patients Without Standard Modifiable Cardiovascular Risk Factors-How Common Are They, and What Are Their Outcomes[END_REF] ). A key goal is to define those individuals at risk of rapid cardiovascular disease progression, in a manner similar to metastatic malignancy, where new agents can be rapidly tested in small sample sizes due to higher event rate. This approach permits elucidation of novel biological mechanisms not currently addressed by existing agents (Figure 1). For instance, recent data suggests that failure of plaque healing might help explain recurrent acute coronary events and might become a new therapeutic target [START_REF] Vergallo | Atherosclerotic Plaque Healing[END_REF] . A consensus definition of those who rapidly progress to have recurrent events for atherosclerosis, heart failure or other disorders, despite best evidence-based therapies may also provide enhanced opportunities to define underlying mechanisms, and to augment the power for targeted therapeutic studies. Similarly, angina or myocardial infarction without obstructive coronary disease, a cause of substantial morbidity with unmet need, might usefully be defined using population level diagnostics for coronary microvascular dysfunction which could then drive rigorous mechanistic studies and improved targeted therapies [START_REF] Camici | Coronary microvascular dysfunction[END_REF] .

Precision medicine and drug development

Common disease states impacting cardiovascular health, driving morbidity and mortality such as atherosclerosis and heart failure, are multi-factorial, and have been shown to consist of many different sub-groups each presumed to be driven by a distinct predominant mechanism (as yet unknown). An extreme example is the syndrome of heart failure which is divided into subgroups based on systolic function-reduced ejection fraction (HFrEF) versus preserved ejection fraction (HFpEF). Interestingly, HFrEF patients, with widely diverse aetiologies-from prior myocardial infarction, to chronic ischemia, to inflammatory and infiltrative conditions, and through to dilated cardiomyopathy of uncertain aetiology, all appear to gain some benefit from the same neurohormonal agents. In contrast, clinical trials of patients with HFpEF have been consistently neutral [START_REF] Yoon | Heart failure with preserved ejection fraction: present status and future directions[END_REF] , up until the recent observation of potential benefit with the SGLT2 inhibitor class of drugs [START_REF] Figtree | Effects of Canagliflozin on Heart Failure Outcomes Associated With Preserved and Reduced Ejection Fraction in Type 2 Diabetes Mellitus[END_REF][START_REF] Kato | Effect of Dapagliflozin on Heart Failure and Mortality in Type 2 Diabetes Mellitus[END_REF] currently under rigorous prospective evaluation in large scale clinical trials [START_REF] Anker | Evaluation of the effects of sodium-glucose co-transporter 2 inhibition with empagliflozin on morbidity and mortality in patients with chronic heart failure and a preserved ejection fraction: rationale for and design of the EMPEROR-Preserved Trial[END_REF] . Our relative failure in this common clinical syndrome may partly relate to the even more broad spectrum of phenotype cluster, which not only includes individuals with bona fide physiologic abnormalities of relaxation e.g. cardiac amyloidosis and restrictive cardiomyopathy, but also an amalgam of conditions where non-cardiac comorbidities and abnormalities contribute, and the only cardiac abnormalities may be changes in load dependent indices of diastolic physiology which may represent a range of disorders from deconditioning to poorly treated hypertension [START_REF] Shah | Precision Medicine for Heart Failure with Preserved Ejection Fraction: An Overview[END_REF][START_REF] Dunlay | Epidemiology of heart failure with preserved ejection fraction[END_REF] . A similarly complex spectrum of "phenotypes" and aetiologies, regarding individual patterns of response to classic risk factors, likely exists for atherosclerosis [START_REF] Kott | Biobanking for discovery of novel cardiovascular biomarkers using imaging-quantified disease burden: protocol for the longitudinal, prospective, BioHEART-CT cohort study[END_REF] . Better stratification by underlying mechanistic markers will be required to make the next advances in the prevention and treatment of these conditions.

Current clinical trials, and subsequently guidelines, of necessity tend to "bundle" patients together based on the aggregated syndromes referred to above. Whilst the evidencebased treatment likely acts on common parts of the pathway, and some pharmaceutical companies would prefer to design validation phases to provide as "wide" a market as possible, this may hinder larger leaps relevant to smaller more etiologically pure sub-groups.

Advances in disease stratification tools (including molecular and advanced imaging technologies) to identify more specific patient populations will help to unravel new mechanisms in these sub-groups, and lead to better targeting of new therapies. Ultimately, these same mechanisms will also be useful to identify potential novel endpoints that will also accelerate subsequent focused clinical trials. In other words, time has perhaps come to improve the balance and give more emphasis to "splitting" rather than "lumping". A more immediate application of genomics to drug development, is the relatively simpler application of pharmacogenomics, where variants at a specific locus are known to alter the metabolism or response to therapeutic agents. There has been mixed support for such an approach from Pharma yet targeting drugs to specific mechanisms in individuals might streamline development pathways with smaller trials and ultimately lead to higher efficacy and safety as well as better outcomes for patients. In the case of clopidogrel, warfarin and statins, drugs already well-established, the literature has become sufficiently strong that guidelines are now available to help implement the use of genetic information to guide treatment with these therapies [START_REF] Johnson | Pharmacogenetics and cardiovascular disease-implications for personalized medicine[END_REF] . The cost effectiveness of implementing these insights has been difficult to prove, with relatively weak benefit on outcomes observed in clinical studies, and so the prospective development and validation of new therapies for patients with specific genotypes has not been broadly undertaken.

Neglected forms of cardiovascular disease contributing to the burden of disease in lowand middle-income countries

It is estimated by the World Health Organization that 75% of the burden of cardiovascular disease across the globe is found in low-and middle-income countries [START_REF] Organization | Projections of mortality and burden of disease[END_REF] . Whilst a major component of this is shared with higher-income countries-in regard to ischemic heart disease, cerebrovascular disease and hypertension, neglected tropical diseases and other infections associated with poverty account for a significant proportion of these cardiovascular categories [START_REF] Celermajer | Cardiovascular disease in the developing world: prevalences, patterns, and the potential of early disease detection[END_REF][START_REF] Moolani | Neglected tropical diseases as hidden causes of cardiovascular disease[END_REF] . It is well recognised that tackling these conditions, such as rheumatic heart disease, endomyocardial fibrosis and Chagas Disease require public health and policy solutions and equitable implementation of existing evidence. However, there are many stages of the disease where new drug therapies are urgently needed, working with governments and clinicians to consider pragmatic issues around delivery and affordability at all stages of development.

Advances in in vitro human models of cardiovascular disease

Pharmacological studies have long been a centrepiece of nonclinical drug evaluation prior to in vivo studies. Well-defined cellular models enabling investigators to screen chemical compound libraries and drug candidates have then been selected for secondary validation screens. Perhaps more than other organs, there is a need for in vitro models that better represent the human heart to understand cellular mechanisms and to test new therapies [START_REF] Ebert | Induced pluripotent stem cells as a disease modeling and drug screening platform[END_REF] .

Several teams around the world have made recent progress on this front by moving from 2D monolayer cultures to 3D spheroids, organoids, or microphysiological systems. This includes both primary cardiac myocytes, or induced pluripotent stem cell-derived cardiomyocytes (iPSC-CMs), along with co-culturing approaches which incorporate endothelial cells [START_REF] Sayed | Clinical trial in a dish using iPSCs shows lovastatin improves endothelial dysfunction and cellular cross-talk in LMNA cardiomyopathy[END_REF] , and cardiac fibroblasts [START_REF] Zhang | Generation of Quiescent Cardiac Fibroblasts From Human Induced Pluripotent Stem Cells for In Vitro Modeling of Cardiac Fibrosis[END_REF] . Such approaches not only enhance the modelling of human heart behaviour, but also prolong the survival of human cardiac myocytes in culture [START_REF] Polonchuk | Cardiac spheroids as promising in vitro models to study the human heart microenvironment[END_REF] . For example, Mei and colleagues have established a model of myocardial infarction in human cardiac organoids that utilises both hypoxia and noradrenaline and recapitulates pathological metabolic, fibrosis, and calcium handling changes at the transcriptomic, structural and functional level [START_REF] Dunlay | Epidemiology of heart failure with preserved ejection fraction[END_REF] . Hudson and team have developed a bioengineered human cardiac organoid platform which provides functional contractile tissue with biological properties similar to native heart tissue. Moreover, the team have applied this model to screen small molecules with pro-regenerative potential, demonstrating that whilst some small molecules were proregenerative, they had the unintended side effect of decreasing contractility [START_REF] Kott | Biobanking for discovery of novel cardiovascular biomarkers using imaging-quantified disease burden: protocol for the longitudinal, prospective, BioHEART-CT cohort study[END_REF] . Similarly, Takeda et al., have demonstrated the potential of 3D cardiac tissue derived from human iPSC-CMs, to screen for drug-induced cardiotoxicity [START_REF] Takeda | Development of In Vitro Drug-Induced Cardiotoxicity Assay by Using Three-Dimensional Cardiac Tissues Derived from Human Induced Pluripotent Stem Cells[END_REF] . Such approaches have the potential to support early drug screening and tackle the biggest gaps in our drug armament, but it will be vital to improve the uniformity of the organoids at the scale necessary for empiric screening, and proof of translatability remains a challenge for all preclinical models in the cardiovascular space. If successful, this will be a major step forward in overcoming the limitations related to our minimal access to the tissues and cells at the centre of the disease.

In vivo translational models

Robust demonstration of both proof-of-mechanism and efficacy in animal disease models is a critical step in drug development that must be achieved prior to progression to human trials.

The cardiovascular field is littered with numerous examples of therapies that show great promise in preclinical studies yet fail during the clinical trial phase. A major challenge for cardiovascular drug development, that is shared by other disease groups, is the failure of many small animal (usually rodent) models to fully reproduce human diseases. For example, rodents are particularly resistant to the development of atherosclerosis, and even more so to the development of unstable plaques. Furthermore, interventions found to be successful in preclinical animal models have been poor at predicting successful intervention in human studies. While the reasons for this are not fully understood, they are at least partly related to fundamental species differences in hemodynamics, lifespan, lipid metabolism (e.g. absence of the cholesteryl ester transfer (CETP) gene in rodents) [START_REF] Ha | Differences in plasma cholesteryl ester transfer activity in sixteen vertebrate species[END_REF] and immune function [START_REF] Zschaler | Differences in innate immune response between man and mouse[END_REF] . Similar difficulties in adequately replicating disease parameters in animal models for other human cardiovascular pathologies, including cardiac arrhythmias (where species differences in cardiac electro-mechanical coupling confound model development [START_REF] Milani-Nejad | Small and large animal models in cardiac contraction research: advantages and disadvantages[END_REF] ), HFpEF, and resistant hypertension, have plagued translation of discovery research into the clinic. Typically, cardiovascular disease models are agnostic to the actual disease mechanisms in humans, not least because there is insufficient information available regarding the biology in humans as a result of the logistical challenges of obtaining human cardiac and vascular disease tissues.

Practices that are routine in clinical research such as a priori sample size and power calculations, randomisation, blinding, and the use of appropriate statistical analyses, were frequently overlooked in preclinical studies, particularly in academic research. Other factors such as clearly defined inclusion and exclusion criteria, littermate controls, full disclosure of all collected data, and confirmation of findings across multiple labs, have historically often been lacking in academia [START_REF] Drucker | Never Waste a Good Crisis: Confronting Reproducibility in Translational Research[END_REF][START_REF] Hausenloy | Translating Cardioprotection for Patient Benefit: The EU-CARDIOPROTECTION COST Action[END_REF][START_REF] Lindsey | Guidelines for measuring cardiac physiology in mice[END_REF] How can the field be improved? First and foremost is the need for rigorous mechanistically faithful models for the indication of interest. Attention to those models which have proven translatability for safety and efficacy in previous drug development programs would also be useful [START_REF] Hausenloy | Translating Cardioprotection for Patient Benefit: The EU-CARDIOPROTECTION COST Action[END_REF] . Where there are no mechanistic models, efforts are likely best focused on human studies to define causal mechanisms. In many instances there is also a need for large animal models, as a bridge between rodent studies and human trials, but here too the need for mechanistic representation cannot be overstated. Finally, closer attention must be paid to best practice in experimental design. In this regard, guidelines such as ARRIVE (Animal Research: Reporting of In Vivo Experiments) [START_REF] Du Sert | The ARRIVE guidelines 2.0: Updated guidelines for reporting animal research[END_REF] and STAIR (Stroke Therapy Academic Industry Roundtable) [START_REF] Savitz | Stroke Treatment Academic Industry Roundtable X: Brain Cytoprotection Therapies in the Reperfusion Era[END_REF] , modelled after the CONSORT (Consolidated Standards for Reporting Trials) statement for clinical research, provide an excellent roadmap. As such, a template flow diagram for reporting animal use and analysis in preclinical studies has been developed [START_REF] Drucker | Never Waste a Good Crisis: Confronting Reproducibility in Translational Research[END_REF] , the Consolidated Standards of Animal Experiment ReporTing (CONSAERT) template.

Choosing the right models and the right endpoints, as well as empirically defining the extent to which the modeled biology and response to intervention translate into humans, should be a quantitative metric for filing with regulatory agencies to move toward clinical studies. This requirement would also enable such data to be made public in a protected manner, thus enabling progressive optimization of the drug development process.

Transition to a staged approach for regulatory evaluation of therapies based on clinically and mechanistically relevant measures of benefit and safety

The high bar that the cardiovascular field has set, with primary endpoints focussed on mortality except when validated surrogates are available are in sharp contrast with oncology in which unvalidated surrogate end points are frequently accepted, creates an unfortunate backlash of disincentivising investment due to high costs, and high risk. Whilst being careful not to lower the ultimate bar, collaborative effort to identify reproducible markers of benefit and safety that could be tested in a staged approach, may be required to ensure efficiency of the cycle, and ongoing investment in the development of cardiovascular drugs. A focus on what is important to the patient, as well as on the biology underlying a more precise disease taxonomy, will be important.

In 2015 the FDA approved alirocumab and evolocumab, monoclonal antibodies to PCSK9, for high risk patients, prior to the large cardiovascular outcome trials being completed [START_REF] Hwang | Temporal Trends and Factors Associated With Cardiovascular Drug Development, 1990 to 2012[END_REF] due to their benefit on lipid profiles in those who were already maximally treated. This is not dissimilar to the pathway that the diabetic field has taken as required by FDA regarding glucose-lowering treatments, and the oncology field with cancer "shrinkage". In each instance there is a substantial tension between such limited approval and the need for demonstration of definitive benefit and avoidance of harm for patients in terms of morbidity and mortality. Innovation in patient-reported or functional endpoints can be accelerated with the arrival of wearables and other new technologies. There is some potential for advances in non-invasive imaging of the myocardium (e.g. post-contrast myocardial T1 mapping for cardiac fibrosis [START_REF] Vo | Society for Cardiovascular Magnetic Resonance (SCMR) expert consensus for CMR imaging endpoints in clinical research: part I -analytical validation and clinical qualification[END_REF] ) and atherosclerosis (e.g. serial CT coronary angiography and plaque characterisation [START_REF] Vaidya | Colchicine Therapy and Plaque Stabilization in Patients With Acute Coronary Syndrome: A CT Coronary Angiography Study[END_REF] ) to enhance the power of early phase trials for relevant drugs, but it will be vital to improve the rigorous mapping of such metrics onto meaningful patient outcomes. The "hard" endpoints required for new agents for management of myocardial infarction, heart failure, or most primary prevention drugs is a consequence of prior adverse outcomes when such putative surrogates were employed. Improving our approaches to defining and testing new endpoints will be a central requirement to advance cardiovascular drug discovery. The advent of real-world tools and large-scale analytics may aid this process, but here again increased information content will likely be necessary to escape the constraints of existing disease nosology and the resultant decades of downstream selection bias. Regulatory agencies have strong track records responding to patient-centred outcomes, and the cardiology community is actively increasing the use of such metrics and novel trial designs to transform the approaches to translational science in the field. A collaborative effort working with regulatory agencies to develop a staged approach may allow for acceleration of efficacious therapies, reducing time and cost, but not rigor and safety.

A key consideration at all phases of research and development of new drug solutions is equity. The exclusion of minority ethnic groups and women, as well as poor representation of low-income countries has been a detriment to the field, and ignores the burden of disease, with ~75-80% of CVD deaths occurring in low and middle-income countries.

There is increasing appreciation of the importance of early engagement with patients and the community in regard to endpoints that are most relevant and important to them. In this regard, it is critical that the cardiovascular sector is proactive in patient and community engagement in the design of studies, and in developing innovative measurement of patient-reported outcomes and experiences. Such patient-reported outcomes can be captured electronically or in person through formats ranging from questionnaires to wearable devices and are used to provide a quantifiable record of the patient's lived experience of a disease.

The inclusion of such measures as trial endpoints allows for a more comprehensive assessment of the burden of disease and the impact of an intervention [START_REF] Calvert | Guidelines for Inclusion of Patient-Reported Outcomes in Clinical Trial Protocols: The SPIRIT-PRO Extension[END_REF] . However, for such endpoints to be accepted as useful indicators regarding the efficacy and. Alue of the intervention by regulatory bodies and professional guideline developers, we need to ensure that the outcomes are validated to reflect underlying pathology and be of prognostic significance clinically, as well as that they are reliable and reproducible.

Innovative clinical trial design

It is likely that disease heterogeneity diminishes the average therapeutic effect. Because of these modest effect sizes, the large-scale, long term follow-up phase III clinical trials needed to demonstrate the effect of interventions on the essential outcomes of CV disease ("hard" cardiovascular endpoints of mortality or combined Major Adverse Cardiovascular Events (MACE)) are expensive and require time for follow-up. Consensus-building between pharmaceutical manufacturers and regulatory bodies, and the inclusion of multiple other stakeholders, to ensure meaningful problems are addressed directly is a key factor for consideration [START_REF] Marquis-Gravel | Improving the Design of Future PCI Trials for Stable Coronary Artery Disease: JACC State-of-the-Art Review[END_REF] . Diversity of the trial population, specifically with appropriate balance of women and different ethnic groups, rural-urban, wealth and education differences, could also be significantly improved [START_REF] Marquis-Gravel | Improving the Design of Future PCI Trials for Stable Coronary Artery Disease: JACC State-of-the-Art Review[END_REF] . At the same time, improved stratification based on precise phenotypes will improve power and success. The estimated cost of a Phase III clinical trial for primary prevention in cardiovascular disease is between $USD250-450 million; approximately 20-times higher than the median estimated cost of Phase III clinical trials overall. Thus, it is possible that improving the efficiency of cardiovascular trials would be an important step to reinvigorate interest, and to accelerate new drug development and translation at all phases of the pipeline.

The advent of electronic health records and potential for routinely collected medical data to populate virtual registries for heart attacks, heart failure and arrhythmia as examples, are one of a number of opportunities to enhance trial design efficiency and lower costs.

Utilization of routinely collected electronic health data will allow self-population of electronic case report form entries to enhance efficiencies of registries and randomized clinical trials. It may also enable the use of additional synthetic or semi-synthetic control arms by matching verum cases with placebo cases from existing patients with the same demographics and baseline characteristics and with an appropriately long enough longitudinal follow-up. Furthermore, the efficient use of electronic medical record data may enhance the integration of the health system with the research and development (R&D) sector, increasing the proportion of patients with specific conditions participating in trials.

The impact of innovation in clinical studies of novel therapies is evident in Astra Zeneca's testing of Farxiga which has been granted Fast Track Designation in the US for heart failure following acute myocardial infarction leveraging a registry-based trial design [START_REF]US for heart failure following acute myocardial infarction leveraging an innovative registry-based trial design[END_REF] .

Inclisiran (Novartis), is a further example where health care system data from the UK's National Health System is being used to dramatically reduce trial costs by rapidly identifying patient populations through administrative data. This powerful collaboration among academic institutions, government and industry will accelerate development as well as potential market access through the National Institute for Health and Care Excellence's (NICE) approval programme and a population-level commercial arrangement to make it widely accessible to patients [START_REF] Iacobucci | Inclisiran: UK to roll out new cholesterol lowering drug from next year[END_REF] . International networks of clinical trials embedded into efficient health systems with such infrastructure would be a major asset.

Important efficiency gains may be derived from conducting trials directly with the patient-clinican dyad rather than through dedicated trial sites or via specific investigators.

This can be further enhanced by utilizing electronic data collection from wearables, and other data collection approaches such as point of care lab tests. Data quality is already dependent on remote monitoring and statistical fraud detection (trained AI algorithms) rather than solely by on-site monitoring visits with source data verification.

Working with key stakeholders towards global solutions

Despite advances in the processes and organisation of clinical guidelines and evidence-based medicine, which would be envisaged to promote the uptake and application of new therapies once proven in large scale Phase III randomised clinical trials, Pharma face a variety of diverse barriers unique to each nation in regard to federal regulatory authorities, and financial reimbursement policies. This requires a dedicated "army" of strategic company staff members who are often engaged in expensive lobbying with little logical relationship between the chance and rate of success and the clinical unmet need or demonstrated health and economic benefit. Whilst country-specific approaches will be required, moving beyond geographic borders will be essential. As healthcare globalizes, patients are increasingly demanding increased access to the latest innovations and are willing to go beyond their current clinical relationships to gain such access. In developing solutions likely to be effective in improving the success of cardiovascular drug development, we should learn from the oncology field where clinical trials are more tightly integrated into clinical care in a much more rapid cycle innovation framework. This will require transformation of the culture of cardiovascular care which often is balkanized on the basis of acuity and revenue for healthcare systems. Such culture change might best be focused on the creation of robust approaches to the implementation of existing proven therapies, which typically are deployed in a highly variable manner even a decade after reaching consensus endorsement by professional society clinical practice guidelines. Resolving these inequities alone would substantially reduce the cost of drug development, long before the advent of novel targets which would be pursued aggressively in tandem.

Community and government perceptions are often inaccurate, with many believing that much of cardiovascular disease is self-inflicted through poor lifestyle choices, or an inevitable part of ageing. This is compounded by the perception that the current armamentarium of drugs for atherosclerosis and heart failure already provide the entire solution. The remaining large gaps in our understanding and targeting of residual risk and missing mechanisms is ever important.

A coordinated strategy engaging patients, clinicians, researchers, regulators and industry needs to be established to focus on the redesign of the cardiovascular drug discovery and development pipeline. Creating a community that provides a common voice to engage all stakeholders including governments, will be a critical first step and such an effort might rationally be convened by collaboration across the relevant professional and patient societies.

Next steps and concluding remarks

The COVID-19 Pandemic has provided the community with a poignant example of an acute health problem in urgent need of solutions. Understanding disease spread and individual susceptibility, as well as discovering new treatments and a vaccine are priorities immediately evident to all. The response of researchers from around the globe, working with health care, government leaders and industry, as well as the broader community, demonstrates the committed, diverse and innovative talent that can be mobilized to address our world's major health challenges. The extent of collaboration with industry, including between companies is also unprecedented 55 , yet the lack of any pre-existent structures within the biomedical ecosystem for global coordination against a pandemic some 100 years after the last example speaks to the core deficits at play when there is not an immediate temporal imperative to manage a disease. It is important to remember that in the 6 months of COVID-19 to date, based on current statistics, ~9 million individuals around the globe will have died of cardiovascular disease, with the majority from low-and middle-income countries. The time is right to continue the momentum and international collaboration achieved to address COVID-19 and commit to taking similar steps and sustaining these in the fight against our greatest killer (Figure 2). Developing a global approach to transform the drug discovery and translation ecosystem for cardiovascular disease, whilst maintaining efforts towards equitable access to established effective treatments, is an imperative not an option.

Figure 1 .

 1 Figure 1. Schematic representation and summary of 5 patient groups with unmet need for

  AI and machine learning approaches are being used to better cluster patient populations and derive more tightly circumscribed patient cohorts. This patient segmentation may derive patient endotypes, but this will likely require a systematic improvement in the information content of phenotypic assessment. Both unsupervised and supervised machine learning approaches are currently being used to better characterize patients with heart failure and those who have had a stroke. Among these efforts are electronic health record-based explorations of several million patients with longitudinal follow-up data (up to 7 years) in a collaboration with the Broad Institute at Harvard and MIT, Boston, MA, US, as well as with Sensyne Health in Oxford, UK. The American Heart Association have also invested substantially in the Precision Medicine Initiative. The resulting datasets offer the opportunity to train models against discrete outcomes and then validate these in prospectively collected data from Randomized Controlled Trials (RCT). Such approaches aim to improve inclusion or exclusion of patients in future clinical trials targeting more distinctly stratified patient cohorts, but ultimately definitive new biology will be required to define new targets.

Figure 2 .

 2 Figure 2. Call to action: collaborative next steps to accelerate discovery, translation and

  . Many cardiovascular models are 'accelerated', for example employing animal models in which the disease phenotype of e.g. atherosclerosis or heart failure develops over weeks rather than months to years, and/or utilising young animals despite the fact that in humans most cardiovascular diseases present during middle age or beyond. Similarly, many therapies are tested in remarkably few genotypes before clinical use, so that relatively rare adverse events may not be detected. Likewise, cardiovascular morbidities seldom present in isolation from other co-morbidities, yet few studies focus on animal models that combine multiple diseases (e.g. superimposing comorbidities such as obesity or hypertension onto the model of interest). Finally, the overwhelming majority of animal studies only use males and thus likely fail to model key disease mechanisms present in females, failing to represent half of the patients for whom these therapies are intended in the real world.

Conflict of Interest Statements

Dr. Figtree reports grants from National Health and Medical Research Council (Australia), personal fees from CSL, and grants from Abbott Diagnostic outside the submitted work. In addition, Dr. Dr. Zannad reports personal fees from Janssen, Novartis, Boston Scientific, Amgen, CVRx, other contributions from cardiorenal, personal fees from AstraZeneca, Vifor Fresenius, Cardior, Cereno pharmacuetical, Applied Therapeutics, Merck, other contributions from CVCT, personal fees from BAYER, Cellprothera, Owkin, Boehringer, Myokardia, Corvidia, and other contributions from G3 pharmaceuticals, outside the submitted work.