I Introduction
Synchronization of multiagent systems (MASs) is an important component of many cooperative control problems, such as rendezvous [1], formation control [2], flocking control [3], containment control [4] and sensor networks [5]. Synchronization problems can be roughly categorized into leaderless and leaderfollowing. In the leaderless synchronization, which is closely related to the consensus problem, the agents aim to reach to a static or dynamic agreement on a common value [6, 7, 8]. On the other hand, in the leaderfollowing synchronization, agents aim to make the agreement on the states generated by a leader. In this paper, we focus on the design of a distributed leaderfollowing algorithm when the only information available about the leader is its sampled state, which is only available to a subset of followers.
Literature review: The leaderfollowing algorithms for single integrator and double integrator dynamics are presented in [9], and for homogeneous LTI systems are proposed in [10] and [11]. For systems constituted of heterogeneous LTI followers, [12] and [13] propose the algorithms to synchronize with a passive zeroinput LTI leader. [14] and [15] develop the controls for the single and double integral system, respectively, to track an active leader (active leader is a leader that has a control input). But their works assume the leader’s control input is available to all the followers. [16] and [17] propose a leaderfollowing algorithm respectively for homogeneous LTI and heterogeneous nonlinear MASs in which the unknown input of the leader is bounded and is not available to any follower. But the control inputs in [16] and [17] have the sliding mode structure and suffer from the wellknown undesirable chattering behavior. We recall that from a practical perspective, chattering is undesirable and leads to excessive control energy expenditure [18]. [19] is the recent result for the leaderfollowing problem, which is based on the result of [16]
and develops a distributed observer to estimate the leader’s state for each follower. Then, the output synchronization of heterogeneous leaderfollower linear systems is achieved by optimal local tracking of the output of the observer. We note that in both
[16] and [19], the active leader is restricted to be linear and have limited input. The work reviewed so far are all converge to leader following in an asymptotic manner, i.e., the settling time to reach an agreement is infinity. For fast convergence, [20],[21] and [22] propose the finitetime synchronization algorithms for single and double integral MASs, where the upper bound of the settling time explicitly depends on the initial state of the MAS. Therefore, to use these algorithms, the centralized knowledge of the initial state of the MAS is essential to estimate the settling time. [23] and [24] propose the fixedtime synchronization algorithms, where the settling time is bounded and independent of the initial state of the MAS. However, for both these finite and fixedtime algorithms, the settling time is upper bounded by a conservative estimation. [25] introduces the specifiedtime synchronization control for the leaderless MASs in which one can determine the settling time exactly in advance. Specifiedtime synchronization can be useful to the applications that require precise acting time, such as target attack at a specified time.Statement of contributions: In this paper, we consider a leaderfollowing problem in which the only information available about the leader is its instantaneous sampled state that is known only to a subset of a group of heterogeneous LTI followers at the sampling times. We make no assumptions about the input of the leader or the structural form of its dynamics. That is, the state of the leader is perceived by the followers as an exogenous signal. The sampled states of the leader can be the states of a physical system (e.g., in a pursuitevasion problem) or a set of desired reference states of a virtual leader (e.g., in a waypoint tracking problem). Given the limited information about the leader, we seek a practical solution that enables the followers to arrive at the sampled state of the leader before the next sampling time. That is, we design a distributed algorithm that steers a group of heterogeneous LTI followers to be at the sampled states of the leader at finite time just before the next sampled state is obtained. We note that practical one step lagged tracking has also been used in [26, 27, 28] for a set of dynamic average consensus algorithms with asymptotic tracking behavior. Our solution is inspired by the minimum energy controller design [29] in the classical optimal control theory, and is proposed for problems where the interaction topology of the followers plus the leader is an acyclic digraph with the leader as the global sink. Directed acyclic interaction topology can be interpreted as the agents only obtaining information from those in front of them (see, [30, 31] for algorithms designed over acyclic graphs). Our algorithm also allows the followers to track the sampled state of the leader with a locally chosen offset, which can be timevarying. This offset, when the followers are mobile agents and their whole state or part of it is the position vector, can be used to enable the followers to form a transnational invariant formation [32] about the sampled state of the leader. For a special class of nonhomogeneous LTI MAS, we show that our results can be extended to solve a leaderfollowing problem where we want only an output of the followers to follow the leader’s sampled state. Finally, we show that if the followers are homogeneous, our algorithm not only results in a leader following behavior, but also it makes the states and inputs of the followers fully synchronized after the first sampling epoch. We demonstrate our leaderfollowing results via three numerical examples. In the first example, we show the application of our leaderfollowing algorithm in following a nonlinear massspringdamper leader under a specific formation structure for a group of heterogeneous linear massspringdamper systems. In the second example, we demonstrate the use of our algorithm for reference state tracking via a group of second order integrator followers with bounded control. The result shows the synchronization of the homogeneous followers is realized. Moreover, using the intrinsic properties of our leader following algorithm, we show that the arrival times at the reference states can be specified in such a way that the inputs of the followers stay within the saturation bounds. Our last example demonstrates an outputtracking scenario for a group of aircraft.
Organization: The rest of this parer is organized as follows. Section II gathers basic notation and graphtheoretic notions. Section III gives our problem definition and objective statement. Section IV proposes our distributed leaderfollowing algorithm. In Section V, three applications are demonstrated. Section VI concludes the results of this paper. Appendix A contains the proof of our main result, Theorem IV.1. Finally, Appendix B presents an auxiliary result, which is invoke to support the feasibility of the sampling time design in our second numerical example.
Ii Notations
Notation: We let , , , , and denote the set of real, positive real, nonnegative real, integer, and nonnegative integer numbers, respectively. The transpose of a matrix is .
Graph theoretic notations and definitions: Here we review our graph related notations and relevant definitions and concepts from graph theory following [33]. A digraph, is a triplet , where is the node set and is the edge set, and is the adjacency matrix of the graph defined according to if and , otherwise. An edge from to means that agent can send information to agent . Here, is called an inneighbor of and is called an outneighbor of . A directed path is a sequence of nodes connected by edges. A directed path that starts and ends at the same node and all other nodes on the path are distinct is called a cycle. A digraph without cycles is called directed acyclic graph. The outdegree of a node is . The outdegree matrix of a graph is . We denote the set of inneighbors of an agent by and the outneighbors of agent by . A node is called a global sink of if it outdegree and for every node there is at least a path from to .
Iii Problem definition
We consider a group of heterogeneous MAS whose dynamics is described by
(1) 
where is the state vector and is the control vector. Throughout the paper we assume that the agents’ dynamics (1) is controllable, i.e., for is controllable. These agents (referred hereafter as followers) aim to follow a dynamic signal with possibly a locally chosen offset. This signal can be a dynamic reference signal of a virtual leader or the state of an active physical leader with (possibly) a nonlinear dynamics, e.g., . The dynamical model and the input of the leader is not known to the followers. The interaction topology between the followers is described by a acyclic digraph, denoted by . Only a subset of followers in , denoted by , has access to at the sampling times , . Throughout the paper we assume that for any with . We let be the digraph consisted of the leader and and the directed edges connecting to the leader. In what follows, we assume that the leader is the global sink of , so that its information reaches all the agents in an explicit or implicit manner (see Fig. 1 for an example). We let be the set of the outneighbors of agent in graph ; we note the . Finally, we call the followers homogeneous if , for .
Give that we only have a limited information about the leader (only the sampled states of the leader is available), we seek a practical solution that enables the followers to arrive at the sampled state of the leader before the next sampling time. Therefore, our objective in this paper is to design a distributed control rule for the input vector of each follower such that
(2) 
That is, the follower can steer itself to be in offset with respect to the state of the leader in time before the next sampling time . We note that the set of offsets , when it is related to the position offsets of the agents, defines the formation of the followers around the leader. Here, the term formation refers to transnational invariant formation [32, Section 6.1.1]. For scenarios where the objective is to synchronize to the state of the leader, is set to zero for all . To form the offset, we assume that at each sampling time , follower knows for ; either the follower is given with respect to its outneighbor or constructs it locally after agent sends its to agent . We note that if the leader is a global sink of , given and a set of , and , we can show that the state offset for follower with respect to the leader is unique.
Iv Main result
In this section, we develop a novel distributed solution to solve the leaderfollowing problem stated in Section III. To present this result, we recall that
(3) 
is the controllability Gramian of for any finite time . Since is controllable, is full rank and invertible at each time . We start by using a classical optimal control result to make the following statement.
Lemma IV.1.
Consider a leaderfollowing with an offset problem where each follower’s dynamics is given by (1) with controllable. Suppose is a follower in that has access to of the leader at each sampling time , , i.e., . Also, is the desired state offset with respect to . Starting at an initial condition with , for any let
(4) 
for , where where , and
(5) 
Then, for every we have for all . Moreover, at each time , the control input of satisfies
(6a)  
(6b)  
(6c) 
Proof.
The proof follows from the classical finite time minimum energy optimal control design [29, page 138]. ∎
Lemma IV.1 essentially states that any follower that samples the leader, in the intersampling time interval can use the classical minimum energy control to steer towards the latest sampled state of the leader (with an offset if specified). Next, we show that this idea can be extended to a distributed setting in which only a subset of the followers have access to the leader’s sampled state. To present our results we first introduce some notations. We denote the adjacency matrix and outdegree matrix of the followers’ interaction topology , respectively, by and . We let
(7) 
be the indicator operator that defines the state of connectivity of follower to the leader. For , we also define
(8a)  
(8b) 
We notice that , where is the controllability Gramian (3). Therefore at each finite time , by virtue of controllability of , is invertible. Moreover, note that using the classical control results we can show that can be computed numerically from where and for are obtained from
With the proper notations at hand, we present our distributed solution to solve our leaderfollowing problem of interest as follows.
Theorem IV.1 (A leaderfollowing algorithm for a group of heterogeneous LTI followers).
Consider a leaderfollowing problem where the followers’ dynamics are given by (1). Suppose the leader’s timevarying state is . Let the network topology be an acyclic digraph with leader, node , as the global sink. Suppose every follower has access to at each sampling time , . Let and be the desired state offset (formation) with reference to and , respectively. Starting at an initial condition with , let for
(9) 
where is given in (8a), , and . Then, the followings hold for and :

, moreover, and ;

the trajectory of every follower is
(10) 
the control input of every agent is equal to (4).
The proof of Theorem IV.1 is given in the appendix. Several observations and remarks are in order regarding the leaderfollowing algorithm of Theorem IV.1.
Remark IV.1 (Robustness to state perturbations).
We observe that the leaderfollowing algorithm of Theorem IV.1 has robustness to state perturbations similar to the wellknown Model Predictive Control (MPC). Even though the controller implemented in each epoch is an openloop control, since every follower exerts its state at time as initial condition to the controller, the algorithm can account for the slight perturbations in the follower final state at the end of each epoch.
Remark IV.2 (Implementation of control law (iv.1)).
To implement (IV.1), we note that the component of (IV.1) that multiplies is computed using the local variables of follower and the sampled state of the leader if is nonzero, i.e., . The component of (IV.1) that multiplies is computed using the local variables of follower and variables of its outneighbors if . To compute this term, if the follower knows of its outneighbor (which is the case e.g., when the group is homogeneous), it can implement control (IV.1) by obtaining the state of its outneighbor and computing and locally. Otherwise, each follower needs to obtain and of its each outneighbor . We note here that since the interval is open from the left and the dynamics of all the followers is controllable, is well defined. However, for from the right, goes to infinity. But, since goes to zero as from the right, the product goes to zero as from the right. The “highgainchallenge” observed here is often a common feature in any approach that is geared towards regulation in prescribed finite time. For example, finitehorizon optimal controls with a terminal constraint inevitably yield gains that go to infinity (see e.g., [34, 35, 36]). In practice, the multiplication of very large and very small values can create numerical problems. To address the problem, one way proposed in the literature is equivalent to employ a deadzone on at the beginning of each time interval. Another approach is equivalent to using a lager interval , where is small positive number, to compute such that for is no longer goes unbounded when goes to from the right. These approaches of course result in somewhat sacrifices in the accuracy at each arrival value at at the end of time interval . However, as discussed in Remark 4.1, the errors will not accumulate. As interestingly discussed in [34], the highgainchallenge in the finitetime control can be contrasted with nonsmooth feedback in the sliding mode control where the gain approaches infinity near sliding surface [37] (but the total control input is zero). However, unlike the sliding mode control, where the practical implementation of the highgain leads to persistent chattering on the sliding surface [37], in our case the concern arises only at the start of each transition from one sampling time to the other. The aforementioned practical measures to handle the highgainchallenge in our setting indeed can be compared to the boundary layer approach [37] in the sliding mode control to eliminate chattering.
Remark IV.3 (Timevarying MAS dynamics and network topology).
From the proof of Theorem IV.1, we can see that the followers dynamics can be allowed to be timevarying but piecewise constant over each time interval , i.e., and , for . Similarly the network topology can be allowed to be timevarying as long as between the topology is fixed and satisfies the connectivity condition of Theorem IV.1.
Remark IV.4 (Minimum energy control in ).
From statement (c) of Theorem IV.1 it follows that at each time interval , , the control input of each follower is the minimum energy controller that transfers the follower from its current state to their desired state .
Remark IV.5 (Tracking a priori known desired states at exact sampling time and design of arrival times).
We note that if the leader is virtual and the sampled states are some desired states that are known a priori to with desired arrival time in , the agents can arrive at the desired state of the leader at the desired arrival time. Furthermore, for the homogeneous followers, in cases that the arrival times is not specified one of the followers in (we refer to it as super node that knows the initial state of all the other followers) can design the arrival times to meet other optimality conditions or to avoid violating constraints such as input saturation. In case of input saturation, the fact that by virtue of statement (c) of Theorem IV.1 the form of input vector of the followers are known to be (4) can be instrumental to the super node in design of arrival times. Our second demonstrative example in Section V offers the details.
Remark IV.6 (Extension of results to output tracking for a special class of MAS).
The design methodology of the state offset (formation) algorithm of Theorem IV.1 can be used in output tracking for a special class of MAS. Let the network topology be as described in Theorem IV.1 and the system dynamics of the followers be (1) where and (the state and input dimensions of the followers are not necessarily the same). Let the objective be that the output , , of each follower should satisfy
(11) 
If is full row rank, we can use the control , , to write the output dynamics of each follower as . Then the method of Theorem IV.1 can be used to design , which can then be used to obtain the appropriate that will make the followers meet (11).
Finally, we note that if the followers are homogeneous, the followers can achieve full synchronization in the sense stated in the result below.
Corollary IV.1 (Full synchronization for homogeneous followers).
Let the state offset be constant i.e., for all or (equivalently for ), and assume that the followers are homogeneous. Then, it follows from statements (b) and (c) of Theorem IV.1 that the followers’ trajectories and inputs satisfy for and for , for every . One can easily verify this point by shifting the state coordinate with . Moreover, if the agents are initially in the specified offset i.e., for all , then these qualities also hold for .
Assume that there exists , for and a controllable pair known to all followers, such that using , , makes the followers dynamic homogeneous, i.e., , and , . Then, it is also possible to achieve full state synchronization by implementing (IV.1) to for heterogeneous followers. One sufficient condition for the existence of and , , is that of each follower is full row rank. Then, , and can be any controllable pair.
V Demonstrative examples
In this section, we demonstrate our results via numerical examples.
Va A nonlinearleader following problem for a group of heterogeneous followers
Consider a group of massspringdamper system (followers)
(12) 
where is the state vector with and representing the displacement and velocity of the mass, , and are spring constant, damping constant and mass, respectively, and is the input force. The system’s parameters for are , , , , , , and , respectively. The leader denoted by is a nonlinear massspringdamper system
(13) 
where the input is unknown to the followers and the system parameters . The interaction topology of the systems is shown in Fig. 1. Followers , and obtain the state of the leader with a sampling rate of per second, i.e., second, . The followers start at , , , , , , in a formation with uniform distance to the previous number of the follower. The objective is for the followers to track the sampled state of the leader while preserving the initial formation of the systems at every sampling time . The follower only knows the local formation, i.e., for . For example, follower 3 knows , , and .
The result of implementing the algorithm of Theorem IV.1 is shown in Fig. 2. The ‘’ represents the sampled leader’s states and ‘’ shows the followers track the sampled state in the desired formation at the next sampled time. In this example interestingly in the transition times similar to what is expected from homogeneous followers the state and input of all the followers are almost offsetsynchronized. However, this property is not necessarily true in general for heterogeneous followers.
VB Reference state tracking for a group of second integrator dynamics with bounded inputs
We consider a group of followers with second order integrator dynamics
(14) 
for . The interaction topology of these followers is shown in Fig. 3, where, agent is the virtual leader that is defined more precisely below. Starting at initial conditions , , , , , , the leaderfollowing mission for this team is to traverse through the sequence of desired states , which for privacy reason are only known to follower . The objective is to meet the sequence of desired states without violating any of the followers’ control bounds. In this problem setting, follower is the super node that knows the initial starting state of all the followers in the team and has computational power to compute the arrival times as follows to meet the team’s objective. First, we note that by virtue of statement (c) of Theorem IV.1 the form of input vector of the followers are known to be (4). Since follower 1 knows for , follower can evaluate of all the followers. Starting with , follower computes the arrival time at desired state from the process below
(15) 
where with . Then, the arrival time so that the followers input do not saturate over is set to . (II) Due to Corollary IV.1, after first epoch, the followers inputs are equal to each other. Then, the remaining arrival time , are computed from the optimization problem
(16) 
where with , for . The solution for this set of sequential optimal control problem is , , and seconds. At the end of process, follower broadcasts the times to the network. Broadcasting the reference states is not allowed due to privacy reasons. We note that the desired arrival times can be done offline by the system operator. To match the notation in (IV.1), at the implantation stage, we set , , and , , where . Figures 4 shows that all the followers meet the desired reference state of the virtual leader at the specified arrival times without delay (the ‘’ marks the reference states). Figure 4 also shows the control history of the agents. As seen, the control inputs respect the saturation bounds or . We can also observe that the followers’ states and inputs, as predicted in Corollary IV.1, are all synchronized after the first epoch. We should mention that by virtue of Lemma B.1 in Appendix B, every , , designed as described above is guaranteed to be a finite value.
VC Output tracking for a group of aircraft
We consider a group of aircraft whose shortperiod dynamics is given by (taken from [38, Example 10.1])
where , and are are respectively, angle of attack, pitch rate and elevator angle of aircraft . The interaction topology of these aircraft is shown in Fig. 3, where, agent is the leader. For this system , therefore the condition of Remark IV.6 is satisfied and we can design a distributed algorithm to synchronize the pitch rate of the follower aircraft to the pitch rate of the leader aircraft when only sampled pitch rate of the leader at every 0.1 seconds is available to the follower aircraft . Figure 5 demonstrates the results.
Vi Conclusion
In this paper, we have proposed a distributed leaderfollowing algorithm for heterogeneous multiagent systems with an active leader with unknown input. We have proved that our distributed leaderfollowing algorithm for the linear followers steers the group to be at the sampled states of the leader at the specified arrival times. We showed that the control input of each follower agent between the sampling times is a minimum energy control. We also showed that after the first sampling epoch, the states of all the homogeneous follower agents are synchronized with each other. We demonstrated our results via leaderfollowing problems of massspringdamper systems, mobile agents with second order integrator dynamics, and a group of aircraft. Future work will focus on extending our results to output following problem.
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