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Adaptive Active-Passive Networked Multiagent Systems

Active-passive multiagent systems consist of agents subject to inputs (active agents) and agents with no inputs (passive agents), where active and passive agent roles are considered to be interchangeable in order to capture a wide array of applications. A challenge in the control of active-passive multiagent systems is the presence of information exchange uncertainties that can yield to undesirable closed-loop system performance. Motivated by this standpoint, this paper proposes an adaptive control algorithm for this class of multiagent systems to suppress the negative effects of information exchange uncertainties. Specifically, by estimating these uncertainties, the proposed adaptive control architecture has the ability to recover the active-passive multiagent system performance in a distributed manner. As a result, the agents converge to a user-adjustable neighborhood of the average of the applied inputs to the active agents. The efficacy of the proposed control architecture is also validated from a human-robot collaboration perspective, where a human is visiting several task locations, and the multiagent system identifies these locations and moves toward them as a coverage control problem.


I. INTRODUCTION
There are considerable amount of studies in the literature focusing on dynamic consensus filters where a multiagent system is subject to observations of a process of interest (i.e., the agents can sense some exogenous inputs) [l-3]. Specifically, these studies consider that each agent/sensor is subject to an input (i.e., the agent can sense a process), where the goal of all agents is to converge to the average of the applied inputs. From a practical standpoint, however, a subset of the agents may not be subject to any inputs. To elucidate this point, consider a camera network as an example, where the goal is to locate the position of an intruder. In this example, only a few cameras in the network can see the intruder (i.e., they are subject to an input) at a given time, where it is required for all agents to converge to the average of the inputs of those few cameras for localization purposes. Hence, the results given in the aforementioned references cannot be utilized. A recently proposed approach called active-passive multiagent systems provides a remedy to this problem [4,5]. This class of multiagent systems consist Tansel Yucelen is an Associate Professor of the Department of Mechanical Engineering and the Director of the Laboratory for Autonomy, Control, Information, and Systems (LACIS) at University of South Florida, Tampa, FL 33620, USA (email: yucelen@usf.edu).
The first and second author would like to acknowledge the support of the Kahn Foundation under award number AWD011321. of agents subject to inputs (active agents) and agents with no inputs (passive agents). Specifically, in [4], an integral action-based distributed control algorithm is proposed, which requires agents to exchange their current state and integral state within each other, under the assumption that active and passive roles of agents are fixed in order to ensure all agents to converge to the average of the inputs of active agents. In [5], the requirements on integral state exchange and fixed active and passive roles are relaxed. Yet, how the presence of information exchange uncertainties can affect the convergence of all agents to the average of the inputs of active agents is largely unknown.
Furthermore, coverage is one of the fundamental objectives in multi-agent systems, either as static coverage [6][7][8] or dynamic coverage [9][10][11][12]. In particular, this problem addresses the optimal placement of sensors to cover a given region and has been extensively studied as the deployment of mobile sensing agents to the centroids of Voronoi cells in a Voronoi partition of a given domain based on a density map, that is usually assumed to be available through local measurements of the process of interest. However, obtaining a density map for converge control design for tracking a target such as in human-robot collaborative networks can be challenging, since the target information is only accessible for a subset of agents, resulting in an active-passive multiagent system as described above.
In this paper, we propose a distributed adaptive control architecture to address the aforementioned challenge in the active-passive multiagent systems in the presence of uncertainties in the communication signals. The proposed control architecture estimates the unknown information exchange uncertainties, and recovers the performance of active-passive multiagent system. Based on a Lyapunov stability analysis, we show that the proposed algorithm drives the agents to a user-adjustable neighborhood of the the average of the applied inputs in the face of these uncertainties. From a human-robot collaboration perspective, we then consider a converge control problem as a case study, where the agents are required to assist a human as it moves within a field. Specifically, considering the human as the only input to this active-passive multiagent system, the proposed adaptive control architecture estimates the position and velocity of the human. The agents then construct a density map and move toward the high density locations via coverage control.

II. MATHEMATICAL PRELIMINARIES
We start by introducing the notation used throughout this paper. lit !Rn, and !Rnxm respectively denote the set of real numbers, the set of n x I real column vectors, and the set of n x m real matrices; On denotes the n x 1 zero vector, ln denotes the n x 1 ones vector, In denotes the n x n identity matrix, and lFt.+ (resp., i"+) and JFt.'.):.xn (resp., i":xn) denote the set of positive real numbers (resp., non-negative reals) and the set of n x n positive-definite (resp., positive semidefinite) real matrices. In addition, we use (-)T to denote the transpose, (-)t to denote generalized inverse, and Amin (A) (resp., Amax (A)) to denote the minimum (resp., maximum) eigenvalue of the square matrix A .
Necessary notions from graph theory are next concisely overviewed (details in [13,14]). Specifically, an undirected graph (vis defined by a set Vo;= {1 , 2, ... , N} of nodes and a set &o; C Vo; x Vo; of edges. The degree of a node is given by the number of its neighbors. Letting di denote the degree of node i, then the degree matrix of a graph m, denoted by The following lemmas are needed for our main results, where we consider that a multiagent system is represented by a connected, undirected graph m. and Yi < 0, and Pro j ( 0, y) ~ Yi otherwise.

MULTIAGENT SYSTEMS
For the main result of this paper presented in the following section, we here concisely overview the active-passive multiagent systems [5,19,20]. Consider N agents exchanging information based on a connected, undirected graph m, where the dynamics of each agent satisfy Here, xi(t) E lFt. is the state of agent i, i = 1, ... ,N, is the control input, and a E JFt., a -=/-0. The agents that can sense the exogenous inputs ch ( t) E JFt., h = 1, ... , m, m ~ N , are called active agents; otherwise, they are called passive agents. As the location of the exogenous input changes with respect to the agents, the active and passive roles of the agents changes (see Figure 1). The objective for the agents is to reach dynamic consensus to a user-adjustable neighborhood of the average of the applied inputs (i.e., the observable process by the active agents). Consider the distributed control algorithm given by i~j where Pi(t) E lFt. denotes the integral action of agent i. In (2) and (3), O'.,"f,a E lFt.+ are design parameters, kih(t) is a smooth function that changes from 1 ( at the location of the agent i) to O (at the sensing radius of the agent i), ko E lFt. is designed such that ao ~ a -ko ~ 0, and /3i E i"+ is designed with at least one nonzero /3i, and let k1 ,i ~ I::i~h kih(t) E i"+ denote the number of the inputs applied to active agent i. We now define such that k1,i(t) ~ I::f= 1 k2,i 1 (t) E i"+ holds. In a compact form, one can now write (1), (2) and (3) as PN(t)JT E ~N, and c(t) ~ [c1(t), ... ,cm(t),0, ... ,0jT E ~N. The average of the applied inputs to the active agents can be given by (7) The following assumption is utilized in studies concerning active-passive multiagent networks.  (9) and use .C(m) = B(m)B(m) T to write (5) and (6) as where F(m) : §: .C(m) + f3 E ~z-xN based on Lemma l. It now follows from (8) that Finally, the closed-loop error dynamics can be expressed as (15) s2(t) ~ -a')'aKc(t)c(t) -aKcc(t) -aKcc(t). (18) Theorem 1 ([5], Theorem 1). The closed-loop error dynam-ics ( 15) and ( 16) are bounded under the distributed control algorithm (2) and (3), and the ultimate upper bound on the error signal 8(t) satisfies ll 8 (t)ll 2 ::; "Ell/3ll(a 2 ll/311 + 2Na°Ed) + N 2°E~ + laolN"E a 2 .X~in(F(m)) (19) where IIBT(m).Ct Theorem 1 shows that the distributed control algorithm by (2) and (3) drives the state of the agents to a neighborhood of the average of the applied inputs. Moreover, by judiciously selection of the design parameters a, 'Y, a and /3i, this neighborhood can be user-adjustable and get arbitrarily small. Yet, the control algorithm in (2) and (3) requires agent i to have access to its own exact states (i.e., Xi ( t)) as well as the exact states of the neighboring agents (i.e., Xj(t), i ~ j). From a practical standpoint, this requirement can be limiting in the presence of sensor and information exchange uncertainties, which can be due to the external disturbances, malicious sensor attacks, sensor failure, sensor bias, or detrimental environmental conditions. Consequently, these uncertainties can significantly deteriorate the closedloop system performance in reaching to the average of the applied inputs in active-passive multiagent system. Next section presents an adaptive control strategy for suppressing the adverse effects of these uncertainties in active-passive multiagent systems.

ARCHITECTURE
In this section, we propose an adaptive control architecture for active-passive multiagent systems subject to information exchange uncertainties. Specifically, consider that each agent has only access to a corrupted measurement of its own state and states of the neighboring agents. Mathematically speaking, the uncertain state measurement of agent i satisfies (20) where b.i(t) E ~ is the time-varying uncertainty of agent i.
For the well-posedness of the considered problem, we now make the following assumption.
Now, consider the distributed adaptive control architecture given by i~j +Pi( t)a I>ih(t) ( Xi (t)ch (t)) + vi (t), (22) where 1\(t) E JR is the estimation of the true integral action of agent i (i.e., the integral action signal when the states are available) having the update law i~j i~j By introducing (22) in (1), one can write the system dynamics in the compact form where x(t) ~ [x1(t),x2(t), ... ,xN(t)jT E !RN. Using (29) and (31), the uncertain state dynamics can be written as
We now substitute the corrective signals from (24)  where s 1(t) and s2(t) are given in (17) and (18), and A(t) ~

~(t) -A(t) E !RN is the uncertainty estimation error.
Next, define ex; (t) ~ xi(t) -xi(t) -Ai(t) ER Note that the compact form of (27) can be written as

■
In the next corollary, we present a special case when the communication uncertainties are constant. (20). Let the adaptive distributed control signal be given by (22) and (23)  Proof. The result directly follows from the proof of Theorem 2; hence, the proof is omitted.

V. ILLUSTRATIVE NUMERICAL EXAMPLE
In this section, we consider a coverage control problem predicated on the proposed adaptive active-passive multiagent system framework to illustrate the efficacy of the proposed approach. In particular, consider an orchard as shown in Figure 2, where a farmer needs to visit several task locations. We assume that a group of mobile sensors are spread throughout the field and are required to assist the human at the task location that are not known for the agents a-priori. Consider that the agents can locally sense the position and the velocity of the human when the human is within their sensing radius R8 • The mobile sensors then infer For this purpose, the proposed adaptive active-passive control architecture is used to estimate the position and the velocity of the human in a distributed manner. Each agent then constructs a density map based on the estimated human position and velocity such that a lower human speed corresponds to a higher density function value. As a result, the obtained time-varying density map represents an importance map for the task locations (see Figure 3). We then utilize the coverage control method introduced in [21] to coordinate the agents based on the time-varying density map toward the task locations.
For this simulation study, we consider 25 agents with the sensing dynamics given in (1)  system. The initial deviation in position estimation during the transient period of the parameter adjustment mechanism can be seen around (0, 0) position in Figure 4.
For the coordination of the mobile sensors, we make use of the coverage control algorithm in [21]. Specifically, a Voronoi tessellation is considered for each mobile sensor This design for the density function increases the relative importance of the points in D when the estimated human velocity is low, that results in indication of the task locations. We now define the cost function The control signal Ui for coordination of sensor i is then calculated by solving the optimization problem [21] minllUill 2 + hl 2 where a is an extended class K, function and ri is a slack variable. In this simulation study, we assume that the connectivity of the network is preserved. However, one may resort to the connectivity maintenance algorithms, such as [22] to include additional constraints for connectivity to the optimization problem. Figure 6 shows the coordination of the mobile sensors based on the constructed density map. Note that, the density map shown in this figure is for agent 1 (initially located as (0, 0)). However, since the estimated human position and velocity for all agents are close to each other based on Figures 4 and 5, the density maps of the other mobile sensors are similar.

VI. CONCLUSION
In this paper, we proposed an adaptive control architecture for active-passive multiagent systems that are subject to information exchange uncertainties. A distributed algorithm was designed to estimation these uncertainties, and to drive the agents to a user-adjustable neighborhood of the average of the applied inputs to the active agents. We illustrated the efficacy of the proposed approach in a coverage control problem to mitigate the adverse effects of information exchange uncertainties within a mobile sensor network.