Distributed adaptive control and stability verification

International Journal of Control

ISSN: 0020-7179 (Print) 1366-5820 (Online) Journal homepage: http://www.tandfonline.com/loi/tcon20

Distributed adaptive control and stability verification for linear multiagent systems with heterogeneous actuator dynamics and system uncertainties K. Merve Dogan, Benjamin C. Gruenwald, Tansel Yucelen, Jonathan A. Muse & Eric A. Butcher To cite this article: K. Merve Dogan, Benjamin C. Gruenwald, Tansel Yucelen, Jonathan A. Muse & Eric A. Butcher (2018): Distributed adaptive control and stability verification for linear multiagent systems with heterogeneous actuator dynamics and system uncertainties, International Journal of Control, DOI: 10.1080/00207179.2018.1454606 To link to this article: https://doi.org/10.1080/00207179.2018.1454606

Accepted author version posted online: 22 Mar 2018. Published online: 05 Apr 2018. Submit your article to this journal

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INTERNATIONAL JOURNAL OF CONTROL,  https://doi.org/./..

Distributed adaptive control and stability verification for linear multiagent systems with heterogeneous actuator dynamics and system uncertainties K. Merve Dogana , Benjamin C. Gruenwalda , Tansel Yucelen

a

, Jonathan A. Museb and Eric A. Butcherc

a

Department of Mechanical Engineering, University of South Florida, Tampa, FL, USA; b Air Force Research Laboratory, Wright-Patterson Air Force Base, Ohio, USA; c Department of Aerospace and Mechanical Engineering, University of Arizona, Tucson, Arizona, USA

ABSTRACT

ARTICLE HISTORY

The contribution of this paper is a control synthesis and stability verification framework for linear timeinvariant multiagent systems with heterogeneous actuator dynamics and system uncertainties. In particular, we first propose a distributed adaptive control architecture in a leader-follower setting for this class of high-order multiagent systems. The proposed architecture uses a hedging method, which alters the ideal reference model dynamics of each agent in order to ensure correct adaptation in the presence of heterogeneous actuator dynamics of these agents. We then use Lyapunov stability theory and linear matrix inequalities to analyse the proposed architecture. This analysis reveals a stability condition, where evaluation of this condition with respect to a given graph topology allows stability verification of the controlled multiagent system. From a practical point of view, this condition also shows a fundamental tradeoff between heterogeneous agent actuation capabilities and unknown parameters in agent dynamics. Several illustrative numerical examples are also provided to demonstrate the efficacy of the proposed architecture.

Received  September  Accepted  March 

1. Introduction 1.1 Literature review Multiagent systems generally consist of groups of agents (e.g. low-cost and small-size unmanned vehicles) that work cooperatively with each other in order to achieve a given set of system objectives. The last two decades have witnessed extensive research studies on these systems owing to their widespread applications in military and civilian environments (e.g. see Olfati-Saber, Fax, & Murray, 2007; Shamma, 2007; Ren & Beard, 2008; Bullo, Cortes, & Martinez, 2009; Mesbahi & Egerstedt, 2010). One of the open problems in the control design for these systems, which is of paramount importance for their practical applications, is the ability of the controlled system to guarantee stability and performance with respect to often nonidentical agent actuation capabilities and unknown parameters in the agent dynamics. In particular, if a multiagent system has agents with nonidentical actuation capabilities (e.g. unmanned vehicles working together with low and high bandwidth actuators), they cannot execute given distributed control approaches identically. As shown by, for example, Dogan, Yucelen, Gruenwald, Muse, and Butcher (2017) in this context and by, for example, Arkin and Egerstedt (2015) and Sakurama, Verriest, and Egerstedt (2015) in the context of temporal heterogeneity, this then can lead to the instability of the controlled multiagent system. Therefore, it is of both theoretical and practical interest to determine the conditions when the controlled multiagent system remains stable in this case. It should be also noted here that these conditions not only depend on how different the actuation capabilities of these agents are, but also the number of, locations of, and CONTACT Tansel Yucelen

[email protected]

©  Informa UK Limited, trading as Taylor & Francis Group

KEYWORDS

Multiagent systems; actuator dynamics; system uncertainty; distributed adaptive control; linear matrix inequalities; stability verification

communication links between these agents with nonidentical actuation capabilities (see Section 3.1 for a motivational example). In addition to heterogeneous actuation between agents, distributed control approaches also suffer from the presence of unknown parameters in agent dynamics. For example, Cheng, Hou, Tan, Lin, and Zhang (2010), Peng, Wang, Zhang, and Sun (2014), Ma et al. (2016), and Wang, Chen, Lin, and Li (2017) (see also references therein) propose intelligent (e.g. neural networks-based) adaptive control approaches and Das and Lewis (2010, 2011), Yucelen and Egerstedt (2012), Zhang and Lewis (2012), Yucelen and Egerstedt (2013), Yucelen and Johnson (2013), De La Torre, Yucelen, and Peterson (2014), Zhang, Feng, Yang, and Liang (2015), Yucelen, Peterson, and Moore (2015), and Sarsilmaz and Yucelen (2018) (see also references therein) propose direct adaptive control approaches to deal with unknown parameters resulting from the lack of excessive modelling efforts and environmental disturbances. Furthermore, for example, Jin, Wang, Yang, and Ye (2017) propose an adaptive control approach that not only deals with such unknown parameters but also considers the uncertainties resulting from gain variations of controllers. While the above adaptive control approaches can preserve the stability of multiagent systems in the presence of unknown parameters, they do not consider the presence of actuator dynamics under an assumption that such dynamics are sufficiently fast as compared with the controlled multiagent system. To summarise, there is a fundamental gap in the design of distributed adaptive controllers for multiagent systems with heterogeneous actuator dynamics and system uncertainties, where

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stability verification steps must be considered in order to rigorously determine the conditions on the stability of the controlled multiagent system. The research purpose of this paper explained in the next subsection is to close this fundamental gap, where we focus on the heterogeneous case (since this case is more general owing to the fact that the homogeneous case is a special case of the heterogeneous case). 1.2 Contribution The main contribution of this paper is a control synthesis and stability verification framework for linear time-invariant multiagent systems with heterogeneous actuator dynamics and system uncertainties. In particular:

r We first propose a distributed adaptive control architecture in a leader–follower setting for this class of high-order multiagent systems. The proposed architecture uses a hedging method, which alters the ideal reference model dynamics of each agent in order to ensure correct adaptation in the presence of actuator dynamics of these agents. r We then use Lyapunov stability theory and linear matrix inequalities to analyse the proposed architecture. Specifically, Lyapunov stability theory is utilised to show each agent error (i.e. the error between an agent dynamics and its hedged reference model dynamics) remains bounded. We then resort to linear matrix inequalities to show the boundedness of the hedged reference model dynamics of each agent, which in turn guarantees that the agent errors asymptotically vanish. It is important to note that our analysis procedure reveals a stability condition discussed next. r Evaluation of the resulting condition with respect to a given graph topology allows stability verification of the controlled multiagent system. From a practical point of view, this condition also shows a fundamental tradeoff between heterogeneous agent actuation capabilities and unknown parameters in agent dynamics. In particular, one can use this condition to assess stability of the controlled multiagent system with respect to, for example, number of leaders and followers in a given multiagent system, location of these leaders and followers on a given graph, and communication links between leaders and followers (i.e. the edges on the given graph). r Finally, illustrative numerical examples are provided to demonstrate the efficacy of the proposed architecture, where we show how the stability of the controlled multiagent system changes with respect to heterogeneity in the actuator dynamics of agents as well as the number of, locations of, and communication links between leaders and followers. While our system-theoretical results are presented in a leader–follower setting, it should be noted here that they can be readily used for many other problems (e.g. consensus, formation, and containment) of multiagent systems. In particular, the proposed control synthesis and stability verification framework have broad applications involving groups of unmanned air, ground, surface, or underwater vehicles for scenarios including cooperative surveillance, reconnaissance, transportation, and

data gathering to name but a few examples, where each possibly low-cost and small-size agent needs to operate in the presence of uncertainties as well as nonidentical actuation capabilities. Note that the proposed architecture of this paper can be viewed as a generalisation of the results in Johnson and Calise (2003a), Gruenwald, Wagner, Yucelen, and Muse (2015), and Gruenwald, Wagner, Yucelen, and Muse (2016a, 2016b), where these results focus on adaptive control synthesis and stability verification for sole uncertain dynamical systems subject to actuator dynamics (i.e. they are not related with multiagent systems). Finally, preliminary conference versions of this paper appeared in Dogan et al. (2017) and Dogan, Gruenwald, Yucelen, Muse, and Butcher (2018), where the first one only considers scalar agent dynamics and the present paper considerably goes beyond the second one by providing detailed proofs of all the results with additional examples and motivation.

2. Notation and definitions The purpose of this section is to introduce the notation used throughout the paper (Section 2.1) and recall some basic definitions from graph theory (e.g. see books Godsil & Royle, 2001; Mesbahi & Egerstedt, 2010 for further details), where we also overview necessary lemmas for our paper (Section 2.2). 2.1 Notation A fairly standard notation is used in this paper. Specifically, R denotes the set of real numbers, Rn denotes the set of n × 1 real column vectors, Rn×m denotes the set of n × m real matrices, n×n n×n (resp., R+ ) R+ denotes the set of positive real numbers, R+ denotes the set of n × n positive-definite (resp., nonnegativedefinite) real matrices, Z denotes the set of integers, Z+ (resp., Z+ ) denotes the set of positive (resp., nonnegative) integers, 0n denotes the n × 1 vector of all zeros, 1n denotes the n × 1 vector of all ones, 0n × n denotes the n × n zero matrix, and In denotes the n × n identity matrix. In addition, we write (·)T for transpose operator, (·)−1 for the inverse operator, and Proj( ·, ·) for the projection operator (Lavretsky & Wise, 2013, Chapter 11.4). Finally, we write λi (A) for the ith eigenvalue of A (A is symmetric and the eigenvalues are ordered from least to greatest value), diag(a) for the diagonal matrix with the vector a on its diagonal, block-diag(·) for the block diagonal matrix, and ‘’ for the equality by definition. 2.2 Graph-theoretical definitions and lemmas Graphs are broadly adopted as a fundamental tool in the multiagent systems literature to encode agent-to-agent interactions. In particular, an undirected graph G is defined by a set VG = {1, . . . , N} of nodes and a set EG ⊂ VG × VG of edges. If (i, j) ∈ EG , then the nodes i and j are neighbours and the neighbouring relation is indicated with i ∼ j. The degree of a node is given by the number of its neighbours. Letting di be the degree of node i, the degree matrix of a graph G, D(G) ∈ RN×N , is defined by D(G)  diag(d), where d = [d1 , … , dN ]T . A path i0 i1 …iL is a finite sequence of nodes such that ik − 1 ∼ ik , k = 1, … , L, and a graph G is connected when there is a path between any pair of distinct nodes. The adjacency matrix of a graph G, A(G) ∈ RN×N , is defined by [A(G)]i j = 1 if (i, j) ∈ EG and [A(G)]i j = N×N

0 otherwise. The Laplacian matrix of a graph, L(G) ∈ R+

,

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playing a central role in many graph theoretic treatments of multiagent systems, is defined by L(G)  D(G) − A(G). Throughout this paper, we model a given multiagent system by a connected, undirected graph G, where nodes and edges, respectively represent agents and inter-agent communication links. In addition, we need the following lemmas, respectively from Mesbahi and Egerstedt (2010), Yucelen and Peterson (2014), and Bernstein (2009). Lemma 2.1: The spectrum of the Laplacian  of aconnected,  undi L(G) < λ rected graph can be ordered as 0 = λ 1 2 L(G) ≤   · · · ≤ λN L(G) with  1nas the eigenvector corresponding to the zero eigenvalue λ1 L(G) and L(G)1N = 0N . Lemma 2.2: Let K = diag(k), k = [k1 , k2 , … , kN ]T , ki ∈ Z+ , i = 1, … , N, and assume at least one element of k is nonzero. Then, for the Laplacian of a connected, undirected graph, F (G)  L(G) + N×N is a positive-definite matrix and −F (G) is Hurwitz. K ∈ R+ Lemma 2.3: Let A1 ∈ Rn×n , A2 ∈ Rn×m , A3 ∈ Rm×n and A4 ∈ Rm×m . If A1 and A4 − A3 A−1 1 A2 are nonsingular, then  −1   A1 A2 M1 M2 , = A3 A4 M3 M4

(1)

−1 −1 where M2  −A−1 1 A2 (A4 − A3 A1 A2 ) .

Note that the result of Lemma 2.2 implicitly depends on the result of Lemma 2.1, where the matrix F (G) of Lemma 2.2 is utilised in the results of this paper. Moreover, the result of Lemma 2.3 is needed for a discussion in the following section.

3. Motivation, problem formulation, and mathematical preliminaries 3.1 Motivational example To elucidate the stability tradeoff between heterogeneous agent actuation capabilities and existing parameters in the dynamics of agents, we first give an example in a simple setting with two scalar agent dynamics given by x˙1 (t ) = u1 (t ) + w1 x1 (t ),

x1 (0) = x10 ,

(2)

x˙2 (t ) = u2 (t ) + w2 x2 (t ),

x2 (0) = x20 ,

(3)

where xi (t ) ∈ R, ui (t ) ∈ R, and wi ∈ R, respectively denote the state, the control input, and a constant of agent i, i = 1, 2. To simplify the control design for now, assume that wi is known and let our objective be to achieve a leader–following behaviour while removing the effect of the term wi xi (t) in (2) and (3). In the absence of actuator dynamics, this problem can be trivially solved with asymptotic stability for any   w i using u1 (t ) = − x1 (t) − x2 (t ) − x1 (t ) − c −w1 x1 (t ) and u2 (t ) = − x2 (t ) − x1 (t ) −w2 x2 (t ) (e.g. see Dogan et al., 2017; Mesbahi & Egerstedt, 2010), where first agent (2) acts as a leader since it has the knowledge of the bounded command c ∈ R to be followed by the states of both agents. In the presence of actuator dynamics, however, one often has the dynamics given by

  v˙ 1 (t ) = −m1 v 1 (t ) − u1 (t ) ,

x˙1 (t ) = v 1 (t ) + w1 x1 (t ), x1 (0) = x10 ,

v 1 (0) = v 10 ,

(4)

  v˙ 2 (t ) = −m2 v 2 (t ) − u2 (t ) ,

x˙2 (t ) = v 2 (t ) + w2 x2 (t ), x2 (0) = x20 ,

3

v 2 (0) = v 20 ,

(5)

where v i (t ) ∈ R and mi denotes the actuator state and the actuator pole of agent i, i = 1, 2. To further simplify the following discussion on the aforementioned tradeoff, set the command c to zero. Based on the control inputs given above, one can then write the resulting closed-loop dynamical system in the compact form given by ⎡

x˙1 (t )

⎤

⎡

w1

1

0

⎥ ⎢ ⎢ ⎢ v˙ 1 (t ) ⎥ ⎢ −m1 (2 + w1 ) −m1 ⎥=⎢ ⎢ ⎥ ⎢ ⎢˙ 0 0 ⎣ x2 (t ) ⎦ ⎣ v˙ 2 (t )

m2



0

m1 w2

0

⎤⎡

x1 (t )

⎤

⎥ ⎥⎢ 0 ⎥ ⎢ v 1 (t ) ⎥ ⎥, ⎥⎢ ⎥ ⎥⎢ 1 ⎦ ⎣ x2 (t ) ⎦

−m2 (1 + w2 ) −m2

 

v 2 (t )

A

⎡

⎤

⎡

⎤

x10 x1 (0) ⎢ ⎥ ⎢ ⎥ ⎢ v 1 (0) ⎥ ⎢ v 10 ⎥ ⎢ ⎥=⎢ ⎥, ⎢ ⎥ ⎢ ⎥ ⎣ x2 (0) ⎦ ⎣ x20 ⎦ v 2 (0)

(6)

v 20

such that A must be Hurwitz for the asymptotic stability of the controlled system. In this simple yet illustrative setting, one can construct, for example, Figure 1 to understand the tradeoff between agent actuation capabilities and unknown parameters in the agent dynamics, where the pair (mi , w i ) changes in homogeneously in the left subfigure and heterogeneously in the right subfigure. For the homogenous case in Figure 1, it is clear that the pole mi of the agent actuators  be large enough for asymp  must totic stability (i.e. max Re λi (A) < 0. For the heterogeneous case, however, we cannot make such a straightforward argument. From a practical standpoint, in addition, it is known that w i can be uncertain, agents of a given multiagent system generally do not obey scalar dynamics, and the number of agents can be finitely many. Thus, the following fundamental research question is immediate: for a given multiagent system having N highorder agents subject to actuator dynamics and system uncertainties, how can we develop a distributed control architecture and a method for performing stability verification of the resulting controlled system? In the remainder of this section, we first introduce the multiagent system set-up considered throughout the paper (Section 3.2). We next concisely overview mathematical preliminaries with regard to basic distributed control methods (Sections 3.3.1 and 3.3.2) focusing on special cases of the presented multiagent system set-up. Note that the main contribution of this paper presented in Section 4 builds on and significantly generalises these basic distributed control methods to rigorously answer the question stated at the end of the previous paragraph.

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m ∈ [0, 1] (m1 = m2 = m)

1

0.9

w ∈ [0, 1] (w1 = w and w2 = 1 − w)

0.9

0.8

w ∈ [0, 1] (w1 = w2 = w)

m ∈ [0, 1] (m1 = m and m2 = 1 − m)

1

0.7

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0 -0.4

-0.2

0

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max Re λi (A)

0.6

0.8

1

-0.2

0

0.2

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max Re λi (A)

0.6

0.8

1

Figure . (Colour online) Change in the real part of the maximum eigenvalue of matrix A in () with respect to the changes in w and m, where w = w  = w and m = m = m in the left subfigure and w  = w, w  =  − w, m = m, and m =  − m in the right subfigure (the line colour variations from blue to red indicate the direction that m    is increased from  to ). Note that stability of the controlled system is guaranteed when max Re λi (A) < 0.

3.2 Multiagent system set-up and distributed control objective Consider a multiagent system consisting of N linear timeinvariant agents with heterogeneous actuator dynamics and system uncertainties. Specifically, the dynamics of agent i, i = 1, … , N, is given by   x˙i (t ) = Axi (t ) + B v i (t ) + WiT xi (t ) ,

xi (0) = xi0 , (7)

yi (t ) = Cxi (t ),

(8)

where A ∈ Rn×n is a known system matrix, B ∈ Rn×m is a known control input matrix, C ∈ R p×n is a known output matrix, and Wi ∈ Rn×m is an unknown weight matrix, which captures the uncertain parameters in the agent dynamics. In addition, xi (t ) ∈ Rn denotes the state vector of agent i, which is only available to this agent and not to its neighbours, yi (t ) ∈ R p denotes the output vector of agent i, which is shared with its neighbours locally through a connected, undirected graph G, and v i (t ) ∈ Rm denotes the output vector of the actuator dynamics of agent i given by x˙ci (t ) = −Mi xci (t ) + ui (t ), v i (t ) = Mi xci (t ),

xci (0) = xci 0 ,

(9) (10)

with xci (t ) ∈ Rm being the actuator state vector, Mi = diag( λ j i ) ∈ Rm×m being a matrix with diagonal entries {λj }i > 0, j = 1, … , m, where these entries represent the actuator bandwidths of each control channel for this agent, and

ui (t ) ∈ Rm being the feedback control input vector. A graphical depiction of the open-loop dynamics (7)–(10) of agent i, i = 1, … , N, is given in Figure 2. For the given multiagent system having N high-order linear time-invariant agents subject to heterogeneous actuator dynamics and system uncertainties, the overarching objective of this paper is to develop a distributed adaptive control architecture and a linear matrix inequalities-based method for performing stability verification such that the agents of the resulting controlled system work in coherence. To address this objective, we utilise a benchmark leader–follower setting in what follows, where the results of this paper can be readily used for many other problems of multiagent systems as discussed above. In this leader–following setting, as standard, we consider that there exists a subset of agents (at least one) that has the knowledge of a given command input to be tracked (i.e. leader agents), where the other agents do not have any knowledge with regard to this command input (i.e. follower agents). Finally, while we consider agents having identical A, B, and C matrices in (7) and (8) for directly focusing on the main results of this paper, the presented contribution can be extended to the case where these matrices are different by resorting to the works in, for example, Sarsilmaz and Yucelen (2017, 2018). In the reminder of this section, we overview necessary mathematical preliminaries with regard to basic distributed control methods (e.g. De La Torre & Yucelen, 2015; De La Torre et al., 2014; Lewis, Zhang, Hengster-Movric, & Das, 2013; Olfati-Saber et al., 2007; Ren & Beard, 2008; Sarsilmaz & Yucelen, 2017, 2018; Tran & Yucelen, 2016; Yucelen & Johnson, 2013) focusing on special cases of the above multiagent system set-up. Specifically, Section 3.3.1 presents a concise review of distributed control

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5

Figure . Open-loop dynamics of agent i, i = , … , N ,given by ()–().

design in a leader–follower setting for the case when there does not exist actuator dynamics and unknown parameters in agent dynamics (i.e. v i (t) = ui (t) and Wi = 0 in (7)). Furthermore, Section 3.3.2 presents a similar concise review for the case when there does not exist actuator dynamics (i.e. v i (t) = ui (t) in (7)) but there are unknown parameters in the agent dynamics. 3.3 Mathematical preliminaries .. Distributed control in the absence of agent actuator dynamics and uncertainties We first present a review of distributed control design in the absence of agent actuator dynamics and uncertainties. In this case, (7)–(10) simplify to x˙i (t ) = Axi (t ) + Bui (t ),

xi (0) = xi0 ,

yi (t ) = Cxi (t ),

(11) (12)

for i = 1, … , N. In order to drive the outputs of all agents to a given command input only available to the leader agents, one can consider the distributed control approach given by ui (t ) = −K1 xi (t ) + K2 zi (t ),     z˙i (t ) = − yi (t ) − y j (t ) −ki yi (t ) − c ,

(13) (14)

i∼ j

where K1 ∈ Rm×n and K2 ∈ Rm×p are feedback controller gain matrices, zi (t ) ∈ R p is an integral state vector, c ∈ R p is a constant command input, and ki  {0, 1} denotes whether the agent is a leader or a follower (i.e. ki = 0 for follower agents and ki = 1 for leader agents). We refer to, for example, Olfati-Saber et al. (2007), Ren and Beard (2008), Lewis et al. (2013), Tran and Yucelen (2016), and Sarsilmaz and Yucelen (2017) for the distributed control approach given by (13) and (14) as well as its similar versions, where the following two assumptions are needed. Assumption 3.1: There exists feedback controller gain matrices K1 and K2 such that  H1i 

 A − BK1 BK2 ∈ R(n+p)×(n+p) , −ρiC 0 p×p

is Hurwitz for all ρ i , i = 1, … , N, where ρi ∈ spec(F (G)).

(15)

Assumption 3.2: There exists K1 and K2 such that J  C(A − BK1 )−1 BK2 ,

(16)

is invertible, where J ∈ R p×p . With Assumptions 3.1 and 3.2, the distributed control approach given by (13) and (14) asymptotically drives the outputs of all agents to a given constant command input only available to leader agents (i.e. limt →  yi (t) = c, i = 1, … , N). To elucidate this point based on the available results in, for example, Tran and Yucelen (2016) and Sarsilmaz and Yucelen (2017), we now provide the following lemma. Lemma 3.1: Consider a multiagent system consisting of N linear time-invariant agents with the dynamics of agent i, i = 1, … , N, given by (11) and (12). In addition, consider the distributed adaptive control architecture in (13) and (14). If Assumptions 3.1 and 3.2 hold, then the distributed control approach given by (13) and (14) asymptotically drives the outputs of all agents to a given constant command input only available to leader agents (i.e. limt →  yi (t) = c, i = 1, … , N). Proof: Based on the results in Tran and Yucelen (2016) and Sarsilmaz and Yucelen (2017), define x(t )  z(t )  [z1 (t ), z2 (t ), . . . , [x1 (t ), x2 (t ), . . . , xN (t )]T ∈ RNn , zN (t )]T ∈ RN p , and y(t )  [y1 (t ), y2 (t ), . . . , yN (t )]T ∈ RN p , and pc  1N ⊗ c ∈ RN p . Now, one can write the dynamics given by ˙ ) = (IN ⊗ (A − BK1 ))x(t ) + (IN ⊗ BK2 )z(t ), x(t ˙ ) = −(F (G) ⊗ C)x(t ) + (K ⊗ I p )pc , z(t

(17) (18)

y(t ) = (IN ⊗ C)x(t ).

(19)

N×N From Lemma 2.2, note that F (G) ∈ R+ is a positive-definite matrix that includes the Laplacian matrix of the considered undirected, connected graph G. In a compact form, (17) and (18) can be rewritten as

¯ ) + Bp ¯ c, ˙ ) = Aq(t q(t

(20)

with q(t )  [xT (t ), zT (t )]T ∈ RN(n+p) and   IN ⊗ (A − BK1 ) IN ⊗ BK2 ¯ ∈ RN(n+p)×N(n+p) , (21) A −F (G) ⊗ C 0 p×p

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  0 B¯  Nn×N p ∈ RN(n+p)×N p . K ⊗ Ip

(22)

  x˙i (t ) = Axi (t ) + B ui (t ) + WiT xi (t ) ,

xi (0) = xi0 , (25)

yi (t ) = Cxi (t ), From Assumption 3.1, note that (21) is Hurwitz, and hence, there exists a unique positiveN(n+p)×N(n+p) such that 0 = definite matrix P¯ ∈ R+ T¯ ¯ ¯ ¯ A P + PA + I holds. Next, consider the Lyapunov ¯ c) = function candidate given by V (q + A¯ −1 Bp −1 ¯ T¯ −1 ¯ ¯ ¯ ¯ (q + A Bpc ) P(q + A Bpc ). Note that A is invertible owing to the fact that it is Hurwitz, and hence, has a nonzero deter¯ c ) > 0 for all q + A¯ −1 Bp ¯ c = 0, minant, V(0) = 0, V (q + A¯ −1 Bp −1 ¯ ¯ and V (q + A Bpc ) is radially unbounded. The timederivative of this Lyapunov function candidate is now given ¯ ¯ c )T (A¯ T P¯ + P¯A)(q(t ¯ c) = ) + A¯ −1 Bp by V˙ (·) = (q(t ) + A¯ −1 Bp  2 −1 ¯ ) < 0. Thus, limt→∞ q(t ) = −A¯ −1 Bp ¯ c. − (q(t ) + A¯ Bp  c 2  N p×N(n+p) −1 ¯ ¯ ¯ Letting C  IN ⊗ C 0N p×N p ∈ R , q = −A Bpc ¯ c at steady-state. Finally, let gives y = −C¯ A¯ −1 Bp   M1 M2 , A¯ −1 = M3 M4

(23)

where we have M2 = −F (G)−1 ⊗ (A − BK1 )−1 BK2 (C(A − BK1 )−1 BK2 )−1 from Lemma 2.3. Note that M2 is well defined ¯ c , we can now owing to Assumption 3.2. From y = −C¯ A¯ −1 Bp write     M1 M2 0N p×N p  p y = − IN ⊗ C 0N p×N p M3 M4 K ⊗ I p c     = − IN ⊗ C M2 K ⊗ I p pc   = IN ⊗ C F (G)−1 ⊗ (A − BK1 )−1  −1   × BK2 C(A − BK1 )−1 BK2 K ⊗ I p pc   = 1N ⊗ I p pc . Therefore, it follows that limt →  yi (t) = c, i = 1, … , N.

(26)

for i = 1, … , N. To suppress the effects of agent uncertainties in a model reference distributed adaptive control setting, we now consider the reference model for agent i, i = 1, … , N, given by x˙ri (t ) = Ar xri (t ) + Br zri (t ), xri (0) = xri 0 ,     yri (t ) − y j (t ) −ki yri (t ) − c , z˙ri (t ) = −

(27) zri (0) = zri 0 ,

i∼ j

(28) yri (t ) = Cxri (t ),

(29)

where xri (t ) ∈ Rn is the reference state vector, zri (t ) ∈ R p is the reference integral state vector, yri (t ) ∈ R p is the reference output vector, and Ar  A − BK1 ∈ Rn×n and Br  BK2 ∈ Rn×p . We refer to Remark 3.3 for details on this reference model selection including why yj (t) is utilised in (28) instead of yr j (t ). In order to drive the outputs of all agents to a given command input only available to the leader agents in the presence of agent uncertainties, one can now consider the distributed adaptive control approach given by ˆ iT (t )xi (t ), ui (t ) = −K1 xi (t ) + K2 zi (t ) − W     z˙i (t ) = − yi (t ) − y j (t ) −ki yi (t ) − c ,

(30) (31)

i∼ j

ˆ i (t ) ∈ Rn×m is an estimate of unknown parameters Wi where W that satisfies the local (i.e. agent-wise) update law (24) 

Remark 3.1: While our above discussions consider the case when the command input is constant, they can be readily extended to the case when the command input is time-varying with a bounded rate (e.g. see Theorem 1 and Corollary 1of Sarsilmaz & Yucelen, 2017 and Section 3 of Tran & Yucelen, 2016). In this case, it is concluded that yi (t) converges to a neighbourhood set of c(t), where the size of this set depends on the command input rate as well as the bandwidth of the controlled system given by (20). Therefore, we consider a constant command input case in what follows for directly focusing on the main results of this paper without loss of much generality. Finally, since the content presented in this subsection based on Assumptions 3.1 and 3.2 is a building block for our main results, these two assumptions are implicitly assumed throughout this paper. .. Distributed control in the absence of agent actuator dynamics and the presence of agent uncertainties We next present a review of distributed control design in the absence of actuator dynamics and the presence of agent uncertainties. In this case, (7)–(10) simplify to

  ˙ˆ T ˆ W i (t ) = α Proj Wi (t ), xi (t )q˜i (t )Pi G ,

(32)

with G  [BT , 0Tm×p ]T ∈ R(n+p)×m , α ∈ R+ being the learning rate gain, and  q˜i (t ) 

     x˜i (t ) x (t ) x (t ) = i − ri ∈ Rn+p , z˜i (t ) zi (t ) zri (t )

(33)

being the augmented agent error state. Considering (32), in ˆ i (t )| ≤ addition, the projection bounds are defined such that |W (n+p)×(n+p) ˆ Wi,max . Moreover, Pi ∈ R+ in (32) satisfies the local T Pi + Pi H2i + I with Lyapunov equation given by 0 = H2i  H2i 

 A − BK1 BK2 ∈ R(n+p)×(n+p) , −(di + ki )C 0 p×p

(34)

when the following assumption holds. Assumption 3.3: There exists K1 and K2 such that H2i is Hurwitz for i = 1, … , N. Remark 3.2: Assumptions 3.1 and 3.3 include the same feedback controller gain matrices K1 and K2 and the same system matrices A, B, and C. Motivated from this fact, one can group

INTERNATIONAL JOURNAL OF CONTROL

these assumptions into a single matrix as Hi 

  A − BK1 BK2 ∈ R(n+p)×(n+p) , −μiC 0 p×p

(35)

where μi takes values from the set containing both ρ i and di + ki ; hence, there exists a positive lower bound μ ∈ R+ and a positive upper bound μ ∈ R+ such that 0 < μ ≤ μi ≤ μ < ∞ since all ρ i and di + ki numbers are positive. That is, when K1 and K2 make Hi Hurwitz for all μi , then Assumptions 3.1 and 3.3 hold. Here, one can resort to methods from robust control analysis or synthesis to, respectively check or ensure Hi is Hurwitz for all μi Olfati-Saber et al. (2007). Specifically, if desired, robust control analysis methods (see, for example Section 2 of Yedavalli, 2016) can be utilised to check whether Hi is Hurwitz or not on the parameter set of μi ∈ [μ, μ] with respect to given K1 and K2 . If Hi is not Hurwitz for all μi , then one can judiciously redesign feedback controller gain matrices K1 and K2 . To perform this design process more systematically, one can also consider adopting methods from robust control synthesis. To this end, several methods (see, for example Section 4 of Yedavalli (2016)) can be utilised to systematically choose the feedback controller gain matrices K1 and K2 which ensure Hi is Hurwitz on the parameter set of μi ∈ [μ, μ]. We refer to, for example, Yucelen and Johnson (2013), De La Torre et al. (2014), De La Torre and Yucelen (2015), and Sarsilmaz and Yucelen (2018) for the distributed adaptive control approach given by (30)–(32) as well as its similar versions, where the following result is based on these references. Lemma 3.2: Consider a multiagent system consisting of N linear time-invariant agents with the dynamics of agent i, i = 1, … , N, given by (25) and (26). In addition, consider the reference model given by (27)–(29) along with the proposed distributed adaptive control architecture in (30)–(32). If Assumptions 3.1–3.3 hold, ˜ (t )) of the closed-loop dynamical sys˜ ), W then the solution (q(t tem is bounded. Proof: From (25), (27), (28), (30), and (31), we write the local error dynamics given by ˜ iT (t )xi (t ), x˙˜i (t ) = Ar x˜i (t ) + Br z˜i (t ) − BW z˜˙i (t ) = −(di + ki )Cx˜i (t ),

(36) (37)

ˆ i (t ) − Wi ∈ Rn×m . Using (33), these local ˜ i (t )  W where W error dynamics can now be written in a compact form given by ˜ iT (t )xi (t ), q˙˜i (t ) = H2i q˜i (t ) − GW

(38)

where notice that H2i is Hurwitz by Assumption 3.3. Next, consider the quadratic function ˜ i, ˜ i ) = q˜Ti Pi q˜i + α −1 trW ˜ iTW Vi (q˜i , W

(39)

˜ i ) > 0 for all (q˜i , W ˜ i ) = and note that V (0, 0) = 0, Vi (q˜i , W ˜ ˜ (0, 0), and Vi (qi , Wi ) is radially unbounded. The time˜ i (t )) = 2q˜Ti (t )Pi [H2i q˜i (t ) − derivative of (39) yields V˙ i (q˜i (t ), W T T ˜ ˜ ˆ i (t ), G Wi (t )xi (t )] + 2 tr Wi (t )Proj [W xi (t ) q˜Ti (t ) Pi G].

7

ˆ i (t ), ˜ iT (t )(Proj(W Using a projection operator property, trW T T xi (t )q˜i (t )Pi G) − xi (t )q˜i (t )Pi G) ≤ 0 (Lavretsky & Wise, ˜ i (t )) ≤ −q˜Ti (t )q˜i (t ) now 2013, Lemma 11.3), V˙ i (q˜i (t ), W follows. Based on (39),  let the Lyapunov function candidate ˜ i ), which yields V˙ (·) ≤ ˜ ) = Ni=1 Vi (q˜i , W ˜ W be given by V (q, T ˜ ). Thus, it is immediate that the pair −q˜ (t )(IN ⊗ In+p )q(t ˜ i (t )) is bounded for all i = 1, … , N. Finally, let xr (t )  (q˜i (t ), W [xr1 (t ), . . . , xrN (t )]T ∈ RNn , zr (t )  [zr1 (t ), . . . , zrN (t )]T ∈ RN p , and qr  [xrT (t ), zrT (t )]T ∈ RN(p+n) . Then, the reference models of all agents can be written in the compact form given ¯ r (t ) + Bp ¯ c + [0Nn×Nn , (A(G) ⊗ C)T ]T x(t ˜ ), by q˙r (t ) = Aq where this expression gives bounded trajectories owing to the facts that A¯ is Hurwitz by Assumption 3.1 and the term ˜ )’ is bounded from the above anal‘[0Nn×Nn , (A(G) ⊗ C)T ]T x(t ysis. From Barbalat’s lemma (Khalil, 1996), V˙ (·) → 0 as t →  is now evident. This result further implies xi (t ) → xri (t ) as t → , zi (t ) → zri (t ) as t → , and yi (t ) → yri (t ) as t → .  Remark 3.3: We use yj (t) in (28) of the reference model for agent i, i = 1, … , N, instead of yr j (t ). This is motivated by our prior works in, for example, De La Torre et al. (2014) and De La Torre and Yucelen (2015), which yields to local update law given by (32) and the aforementioned local Lyapunov equation. In this setting, in addition, the reference model given by (27)– (29) does not require any additional information from neighbouring agents such as the output of their reference models yr j (t ). Note that this does not prevent the outputs of all agents to converge to a given constant command input only available to leader agents. To elucidate this point, notice that since the term ˜ )’ asymptotically goes to zero in ‘[0Nn×Nn , (A(G) ⊗ C)T ]T x(t ¯ r (t ) + Bp ¯ c + [0Nn×Nn , (A(G) ⊗ C)T ]T x(t ˜ ), one can q˙r (t ) = Aq ˙¯ ) = still recover the ideal reference model dynamics given by q(t ¯ ¯ ¯ ) + Bpc , which can be given in the local form for i = 1, … , Aq(t N as x˙ridi (t ) = Ar xridi (t ) + Br zridi (t ), xridi (0) = xridi 0 ,     z˙ridi (t ) = − yridi (t ) − yridj (t ) −ki yridi (t ) − c ,

(40) zridi (0)

= zridi 0 ,

i∼ j

(41) yridi (t ) = Cxridi (t ),

(42)

where xridi (t ) ∈ Rn is the ideal reference state vector, zridi (t ) ∈ R p is the ideal reference integral state vector, yridi (t ) ∈ R p is the ideal  ¯ ) = (xrid1 (t ))T , . . . , (xridN (t ))T , reference output vector. Here, q(t T (zrid1 (t ))T , . . . , (zridN (t ))T . Since limt→∞ yridi (t ) = c by Lemma 3.1 subject to Assumptions 3.1 and 3.2 (notice that (40)– (42) are dynamically identical to (11) subject to (13), (14), and (12) of Section 3.3.1, respectively), it now follows from the above discussion that limt →  yi (t) = c. Finally, similar to the last part of the discussion in Remark 3.1, since the contents presented in this subsection and Section 3.3.1 are based on Assumptions 3.1–3.3 and since these contents are building bricks for our main results presented in the next section, we make these three assumptions implicitly throughout this paper.

8

K. M. DOGAN ET AL.

4. Distributed adaptive control and stability verification In this section, we present a distributed adaptive control architecture and a method for performing stability verification based on the multiagent system set-up introduced in Section 3.2, where each agent can be subject to heterogeneous actuation capabilities and unknown parameters (see Figure 2). While the preliminary contents presented in Sections 3.3.1 and 3.3.2 are the building bricks of our theoretical development here, these results do not consider actuator dynamics. Thus, they can yield to unstable controlled system trajectories when they are directly considered for the multiagent system set-up in Section 3.2 (e.g. see Figure 1 for a motivational simple example). To this end, we utilise linear matrix inequalities in this section in order to reveal the fundamental tradeoff between heterogeneous agent actuation capabilities and unknown parameters in agent dynamics, and hence, the proposed approach allows for stable distributed adaptation. We start by arranging (7) as   x˙i (t ) = Axi (t ) + B v i (t ) + WiT xi (t )   = Axi (t ) + B ui (t ) + WiT xi (t ) + B [v i (t ) − ui (t )] . (43) Using the distributed adaptive control approach in (30)–(32), we now rewrite (43) as ˜ iT (t )xi (t ) + B [v i (t ) − ui (t )] . x˙i (t ) = Ar xi (t ) + Br zi (t ) − BW (44) To achieve correct local adaptation that is not affected by the presence of heterogeneous agent actuator dynamics, we next consider a hedging method-based reference model given by x˙ri (t ) = Ar xri (t ) + Br zri (t ) + B [v i (t ) − ui (t )] , xri (0) = xri 0 , z˙ri (t ) = −

    yri (t ) − y j (t ) −ki yri (t ) − c ,

(45) zri (0) = zri 0 ,

i∼ j

yri (t ) = Cxri (t ),

(46) (47)

instead of the one introduced in (27)–(29), where ‘B[v i (t) − ui (t)]’ in (45) is the hedging term. Specifically, since not all agent reference models can be followed in the presence of actuator dynamics, this method adds ‘B[v i (t) − ui (t)]’ to the reference models for i = 1, … , N such that the resulting reference model trajectories can be followed by the agent dynamics in the presence of system uncertainties (we refer to Johnson, 2000; Johnson & Calise, 2003a, 2003b for additional details). Now, with the hedging term added to the reference model, one can write the same agent error dynamics in Section 3.3.1 as ˜ iT (t )xi (t ), x˙˜i (t ) = Ar x˜i (t ) + Br z˜i (t ) − BW z˙˜i (t ) = −(di + ki )Cx˜i (t ),

(48) (49)

which can be written in a compact form given by ˜ iT (t )xi (t ). q˙˜i (t ) = H2i q˜i (t ) − GW

(50)

The following stability condition is needed for the results of this paper. Assumption 4.1: The square matrix given by ⎡

⎤ IN ⊗ A + φ1 0Nn×N p B ˆ i (t ), Mi ) = ⎣ −F (G) ⊗ C 0N p×N p 0N p×Nm ⎦ A(W −IN ⊗ K1 − φ2 IN ⊗ K2 −M ∈ RN(n+p+m)×N(n+p+m) ,

(51)

is quadratically stable such that there exists a positive-definite matrix P ∈ RN(n+p+m)×N(n+p+m) satisfying the Lyapunov inequality given by ˆ i (t ), Mi )P + PA(W ˆ i (t ), Mi ) < 0, AT (W

(52)

ˆ 1T , . . . , BW ˆ NT ) ∈ RNn×Nn , φ2  where φ1  block-diag(BW T T Nm×Nn ˆ ˆ , B  block-diag(BM1 , block-diag(W1 , . . . , WN ) ∈ R . . . , BMN ) ∈ RNn×Nm , and M  block-diag(M1 , . . . , MN ) ∈ RNm×Nm . The following theorem, which presents the main result of this paper, addresses the distributed control objective highlighted in the second paragraph of Section 3.2. Theorem 4.1: Consider a multiagent system consisting of N linear time-invariant agents with the dynamics of agent i, i = 1, … , N, given by (7) and (8) subject to the actuator dynamics in (9) and (10). In addition, consider the proposed hedging method-based reference model given by (45)–(47) along with the proposed distributed adaptive control architecture in (30)– (32). If Assumptions 3.1–3.3 and 4.1 hold, then the solution ˜ (t ), xr (t ), zr (t ), v (t )) of the closed-loop dynamical sys˜ ), W (q(t ˜ ) = 0. tem is bounded and limt→∞ q(t ˜ (t )) ˜ ), W Proof: We first show the boundedness of the pair (q(t based on the results in Lemma 3.2. Specifically, considering the quadratic function given by (39), one can write ˜ i (t )) ≤ −q˜Ti (t )q˜i (t ). Now, based on the Lyapunov V˙ i (q˜i (t ), W  ˜ i ), one ˜ ) = Ni=1 Vi (q˜i , W ˜ W function candidate given by V (q, ˜ ) holds, and can readily see that V˙ (·) ≤ −q˜T (t )(IN ⊗ In+p )q(t hence, this conclusion yields the boundedness of the pair ˜ (t )). Unlike the results in Lemma 3.2, however, one ˜ ), W (q(t needs to ensure the boundedness of the triple (xr (t), zr (t), v(t)) to utilise the Barbalat’s lemma here. In particular, to show the boundedness of this triple, the dynamics of the agent reference models given by (45) and (46) and the dynamics of the agent actuators given by (9) and (10) can be rearranged using the distributed adaptive control approach in (30) and (31) as x˙ri (t ) = Ar xri (t ) + Br zri (t ) + B[Mi xci (t ) + K1 xi (t ) ˆ iT (t )xi (t )] − K2 zi (t ) + W ˆ iT (t )xri (t ) + BK1 x˜i (t ) = Axri (t ) + BMi xci (t ) + BW ˆ iT (t )x˜i (t ), (53) − BK2 z˜i (t ) + BW

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z˙ri (t ) = −

˜ ) = 0, note Finally, in order to conclude that limt→∞ q(t ˜ ), that q(t) is bounded as a result of the boundedness of q(t ˙ ˜ ) is bounded, xr (t), and zr . It now follows from (50) that q(t and hence, V¨ (·) is bounded for all t ∈ R+ . It then follows from Barbalat’s lemma that limt→∞ V˙ (·) = 0, which conse˜ ) → 0 as t → . The proof is now quently shows that q(t complete. 

    yri (t ) − y j (t ) −ki yri (t ) − c i∼ j

    yri (t ) − yr j (t ) −ki yri (t ) − c =− i∼ j

  yr j (t ) − y j (t ) − j∈N

      yri (t ) − yr j (t ) −ki yri (t ) − c − Cx˜ j , =− i∼ j

9

j∈N

(54)

ˆ iT (t )xi (t ) x˙ci (t ) = −Mi xci (t ) − K1 xi (t ) + K2 zi (t ) − W ˆ iT (t )xri (t ) = −Mi xci (t ) − K1 xri (t ) + K2 zri (t ) − W ˆ iT (t )x˜i (t ). − K1 x˜i (t ) + K2 z˜i (t ) − W

(55)

ˆ i (t ), Mi ) The quadratic stability condition on the matrix A(W stated in Assumption 4.1 and utilised in Theorem 4.1 is important in order to reveal the fundamental tradeoff between heterogeneous agent actuation capabilities and unknown parameters in the agent dynamics (see also Section 5). Yet, we need the following remark that highlights how we can numerically evaluate this quadratic stability condition using linear matrix inequalities.

Remark 4.1: We now utilise linear matrix inequalities to satisfy ˜ ) = the quadratic stability of (51) for given projection bounds W Now, let the aggregated vectors be given by x(t ˆ i,max ˜ ) = [z˜1 (t ), z˜2 (t ), . . . , and the bandwidths of the actuator dynamics M . For this purz(t [x˜1 (t ), x˜2 (t ), . . . , x˜N (t )]T ∈ RNn , i xr (t ) = [xr1 (t ), xr2 (t ), . . . , xrN (t )]T ∈ pose, let W z˜N (t )]T ∈ RN p , n×m ∈ R be defined as i j ,..., j 1 l RNn , zr (t ) = [zr1 (t ), zr2 (t ), . . . , zrN (t )]T ∈ RN p , xc (t ) = T Nm [xc1 (t ), xc2 (t ), . . . , xcN (t )] ∈ R , such that (53)–(55) can be compactly written in the form given by ⎤ ⎡ ˆ i,max,1 (−1) j1+n W ˆ i,max,1+n . . . (−1) j1+(m−1)n W ˆ i,max,1+(m−1)n (−1) j1 W ⎥ ⎢ ˆ i,max,2 (−1) j2+n W ˆ i,max,2+n . . . (−1) j2+(m−1)n W ˆ i,max,2+(m−1)n ⎥ ⎢ (−1) j2 W ⎥ ⎢ (61) W i j1 ,..., jl = ⎢ ⎥, .. .. .. .. ⎥ ⎢ . . . . ⎦ ⎣ ˆ i,max,n (−1) j2n W ˆ i,max,2n . . . ˆ i,max,mn (−1) jn W (−1) jmn W ˜ ) x˙r (t ) = (IN ⊗ A + φ1 )xr (t ) + Bxc (t ) + (IN ⊗ BK1 )x(t ˜ ), ˜ ) + φ1 x(t (56) − (IN ⊗ BK2 )z(t z˙r (t ) = −(F (G) ⊗ C)xr (t ) + (K ⊗ I p )pc , (57) x˙c (t ) = −Mxc (t ) − (IN ⊗ K1 )xr (t ) − φ2 xr (t ) ˜ ) ˜ ) − φ2 x(t + (IN ⊗ K2 )zr (t ) − (IN ⊗ K1 )x(t ˜ ). +(IN ⊗ K2 )z(t Next, defining ξ (t )  written as

[xrT (t ), zrT (t ), xcT (t )]T ,

(58)

(56)–(58) can be

ˆ (t ), M)ξ (t ) + ω(·), ξ˙(t ) = A(W

for agents i = 1, … , N, where jl  {1, 2}, l  {1, … , 2mn } such that W i j1 ,..., jl represents the corners of the hypercube defining the ˆ i (t ), for each agent i = 1, … , N. Furmaximum variation of W ˆ 1T , . . . , BW ˆ NT ) thermore, since we define φ1  block-diag(BW ˆ 1T , . . . , W ˆ NT ), let φ 1 and φ2  block-diag(W ∈ RNn×Nn and 1,...,r Nm×Nn φ 21,...,r ∈ R be defined respectively as

(59)

where ω(·) satisfies ⎡

⎤ ˜ ) ˜ ) − (IN ⊗ BK2 )z(t ˜ ) + φ1 x(t (IN ⊗ BK1 )x(t ⎦. (K ⊗ Ip )pc ω(·) = ⎣ ˜ ) + (IN ⊗ K2 )z(t ˜ ) − φ2 x(t ˜ ) −(IN ⊗ K1 )x(t (60) Notice that ω(·) is a bounded perturbation as a result of Lya˜ ). It now follows that since ω(·) is ˜ W punov stability of (q, ˆ bounded and A(W (t ), M) is quadratically stable by Assumption 4.1, then xr (t), zr (t), and xc (t) are bounded (see, for example, Khalil, 1996). This further implies the boundedness of the actuator output v(t), and hence, the boundedness of the triple (xr (t), zr (t), v(t)).

⎡

T

BW 1 j ,..., j

0n×n

l ⎢ ⎢ ⎢ 0n×n BW T2 j1 ,..., jl φ 11,...,r  ⎢ ⎢ .. .. ⎢ . . ⎣ 0n×n ... ⎡ T W 0m×n ⎢ 1 j1 ,..., jl ⎢ ⎢ 0m×n W T2 j1 ,..., jl φ 21,...,r  ⎢ ⎢ . .. ⎢ .. . ⎣ 1

0m×n

...

... .. . .. .

0n×n .. .

⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦

0n×n T 0n×n BW N j ,..., j 1 ⎤l . . . 0m×n ⎥ .. ⎥ .. ⎥ . . ⎥, ⎥ .. . 0m×n ⎥ ⎦ T 0m×n W N j ,..., j 1

(62)

(63)

l

where r  {1, … , (2mn )N }. Following the results in Muse (2014) and Boyd, Ghaoui, Feron, and Balakrishnan (1994), if the square matrix given by ⎡

A1,...,r

⎤ IN ⊗ A + φ 11,...,r 0Nn×N p B 0N p×N p 0N p×Nm ⎦ , = ⎣ −F (G) ⊗ C −IN ⊗ K1 − φ 21,...,r IN ⊗ K2 −M

(64)

10

K. M. DOGAN ET AL.

each agent. In the illustrative numerical section, this is shown next for several different multiagent formations to investigate how the linear matrix inequalities calculated feasible stability region changes.

Figure . Star graph for Example  (Example . on the left subfigure and Example . on the right subfigure) with ‘dark grey’ colour indicating the leader and ‘light grey’ colour indicating the followers.

5. Illustrative numerical examples In order to illustrate the proposed distributed adaptive control architecture for linear time-invariant multiagent systems with heterogeneous actuator dynamics and system uncertainties, we consider a group of agents with the dynamics: 

   0 1 0 x˙i (t ) = x (t ) + (v i (t ) + WiT xi (t )), 0.5 0.5 i 1

satisfies the linear matrix inequality AT1,...,r P + PA1,...,r < 0,

P > 0,

xi (0) = [0, 0]T ,

(65)

for all permutations of φ 11,...,r and φ 21,...,r , then (51) is quadratically stable. We can then cast (65) as a convex optimisation problem and solve it using linear matrix inequalities for stability verification purposes. As a result of Theorem 4.1 and the use of linear matrix inequalities highlighted in Remark 4.1, for a given multiagent system with heterogeneous actuator dynamics and internal system uncertainties, a feasible stability region can be obtained to determine appropriate operating actuator bandwidth values for

(66)

 yi (t ) = 1

 1 xi (t ).

(67)

For each agent, we choose the reference model given by (45) with the matrices 

 0 1 , −0.5 −1

Ar =

 Br =

 0 , 0.5

(68)

λ2

30 Ex.1.1 LMI Limit Ex.1.2 LMI Limit (λ1 ,λ2 )=(50,27.02)

20 10 35

40

45

50

λ1

55

60

70

Ex.1.1 Simulated Limit with c Ex.1.2 Simulated Limit with c (λ1 ,λ2 )=(50,17.46)

30

λ2

65

20 10 20

25

30

35

40

45

λ1

50

60

65

70

Ex.1.1 Simulated Limit with c(t) Ex.1.2 Simulated Limit with c(t) (λ1 ,λ2 )=(50,16.51)

30

λ2

55

20 10 20

25

30

35

40

45

λ1

50

55

60

65

70

Figure . (Colour online) Feasible region of actuator bandwidth values for Example . The top subplot presents the linear matrix inequalities calculated region, the middle subplot shows the simulated points yielding unbounded reference model trajectories with c = , and the bottom subplot shows the simulated points yielding unbounded reference model trajectories with c(t) = . sin (. π t).

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c x1 (t) xr1 (t)

0.5

0

0

0.5

0

100 200 t (sec)

1

x2 (t) xr2 (t)

0

0

x3 (t) xr3 (t)

0

100 200 t (sec)

1 xi (t), xri (t)

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Figure . Line graphs for Example  (Example . on the top subfigure, Example . in the middle subfigure, and Example . on the bottom subfigure) with ‘dark grey’ colour indicating the leader and ‘light grey’ colour indicating the followers.

which yields the nominal control gains K1 = [1, 1.5] and K2 = 0.5. In the following examples, all agents have a single channel actuator for executing their control input such that Mi = λi , λi ∈ R+ . In addition, we set Wi = [0.45; 0.45] and use ˆ i (t )]1,1 | ≤ 0.5 and |[W ˆ i (t )]2,1 | ≤ 0.5 as the bounds of the |[W (rectangular) projection operator. In what follows, we consider nonidentical multiagent graph topologies to investigate the effect of leader location, additional information links, and ratio of leaders to followers. In each case, we consider the leaders (‘dark grey’ in respective figures) to have an actuator bandwidth λ1 and the followers (‘light grey’ in respective figures) to have an actuator bandwidth λ2 . Using the projection bounds in the linear matrix inequality-based stability verification analysis outlined in Remark 4.1, the feasible region of allowable actuator bandwidth limits for λ1 and λ2 is calculated. As a result, appropriate operating actuator bandwidth values for the leader and follower agents are obtained. Example 1 (Effect of leader location on a star graph): In this example, we consider a star graph topology given in Figure 3. The effect of the leader location is investigated by comparing two cases. For the first case (Example 1.1), we place the leader at one

of the corners of the star graph and in the second case (Example 1.2), we place the leader in the centre of the star graph. A linear matrix inequality analysis is then conducted for both configurations of the leader location in the star graph. To this end, Figure 4 provides a comparison of the resulting feasible stability regions of allowable actuator bandwidth values. Specifically, the top subplot on this figure shows the linear matrix inequality results such that when the pair (λ1 , λ2 ) is selected from the region staying above these lines, then the controlled multiagent system is guaranteed to be stable. On the same figure, the middle and bottom subplots show the minimum unstable values of the pair (λ1 , λ2 ) obtained numerically from the simulation with c = 1 and c(t) = 0.5 sin (0.01 π t), respectively.1 Comparing the results of middle and bottom subplots with the one on top subplot, it can be readily seen that the proposed linear matrix inequality results provide less conservative conclusions on determining the feasible stability regions. Furthermore, it is clear that the change in the location of the leader significantly alters the resulting feasible stability regions. To show the performance of the proposed distributed adaptive control design, three points are selected from Figure 4 for the case of Example 1.1, which are indicated by the black circles. The numerical simulation results are shown at these actuator bandwidth values in Figures 5 and 6 with the constant command c = 1 and Figures 7 and 8 with the bounded time-varying command c(t) = 0.5 sin (0.01 π t), respectively. In particular, since the feasible linear matrix inequality boundary corresponds to calculated minimum feasible values for the actuator bandwidths, it is expected that the system performances are guaranteed to be stable for actuator dynamics at points greater than or equal to the calculated feasible values on the pair (λ1 , λ2 ) as can be seen in Figures 5 and 7. It is not until the actuator bandwidth values are decreased below the linear matrix inequalities calculated limit that the multiagent system reaches a point of instability due to the unbounded reference model trajectory as seen in Figures 6 and 8. Note that we choose to show only three points of only one configuration of the leader location to avoid repetitiveness as the

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Ex.2.1 LMI Limit Ex.2.2 LMI Limit Ex.2.3 LMI Limit (λ1 ,λ2 )=(30,15.24)

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other actuator bandwidth value pairs shown in Figure 4 would provide similar simulation plots as in Figures 5–8. Example 2 (Effect of leader location on a line graph): In this example, we consider a line graph topology with nonidentical leader locations given in Figure 9. The effect of the leader location is investigated by comparing three cases. For the first case (Example 2.1), we place the leader in the middle of the line graph and for the remaining two cases we move the leader toward the front of the line graph (Examples 2.2 and 2.3, respectively). We then conducted a linear matrix inequality analysis for these

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three configurations. Specifically, the top subplot of Figure 10 presents the resulting linear matrix inequality feasible stability regions of allowable actuator bandwidth values and the bottom subplot of the same figure shows the minimum unstable values obtained numerically from the simulation for a given constant command c = 1. Consistent with the first example, it can be seen that the change in the location of the leader significantly alters the resulting feasible stability regions. Finally, to show the performance of the proposed distributed adaptive control design, two points are selected from Figure 10 for the case of Example

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show the performance of the proposed distributed adaptive control design, two points are selected from Figure 14 for the case of Example 3.2, which are indicated by the black circles. The numerical simulation results are shown at these actuator bandwidth values in Figures 15 and 16 with command c = 1, where similar conclusions are observed as for the case in Examples 1 and 2.

Figure . Circle graphs for Example  (Example . on the left subfigure and Example . on the right subfigure) with ‘dark grey’colour indicating the leader and ‘light grey’ colour indicating the followers.

2.1, which are indicated by the black circles. The numerical simulation results are shown at these actuator bandwidth values in Figures 11 and 12 with the command c = 1, where a similar conclusion is observed as for the case in Example 1. Example 3 (Effect of leader location on a circle graph): In this example, we consider a circle graph topology with nonidentical leader locations given in Figure 13. The effect of leader location is investigated by comparing two cases. These cases now include two leaders. For the first case (Example 3.1), we place the leaders as separated by followers such that they do not directly exchange information with each other, and in the second case (Example 3.2) we place the leaders next to each other such that they do directly exchange information. The feasible region of allowable actuator bandwidth pairs is shown in Figure 14 for the two different leader locations in the circle graph. We then perform a linear matrix inequality analysis for these two configurations. In particular, the top subplot of Figure 14 presents the resulting linear matrix inequality feasible stability regions of allowable actuator bandwidth values and the bottom subplot of the same figure shows the minimum unstable values obtained numerically from the simulation for a given constant command c = 1. Once again, the change in the locations of two leaders results in different feasible stability regions. Finally, to

Example 4 (Number of the information links on a circle graph): In this example, we consider a circle graph topology with added information links between agents; see in Figure 17. The effect of the added information links is investigated by comparing three cases. For the first case (Example 4.1), the leader shares information with two followers, for the second case (Example 4.2) the leader shares information with three followers, and for the last case (Example 4.3) all the agents share information with each other. We then conducted a linear matrix inequality analysis for these three configurations. Specifically, the top subplot of Figure 18 presents the resulting linear matrix inequality feasible stability regions of allowable actuator bandwidth values and the bottom subplot of the same figure shows the minimum unstable values obtained numerically from the simulation for a given constant command c = 1. Comparing the result of bottom subplot with the top subplot, it is obvious that the proposed linear matrix inequality results provide less conservative conclusions on determining the feasible stability regions. Furthermore, it is clear that the change in the number of the information links significantly alters the resulting feasible stability regions. In particular, one can conclude from Figure 18 that as information links are added, the feasible stability region shifts upwards; hence, it decreases with the added information links for the considered range of the pair (λ1 , λ2 ). Finally, to show the performance of the proposed distributed adaptive control design, two points are selected from Figure 18 for the case of Example 4.1, which are indicated by black circles. The numerical simulation results are shown at these actuator bandwidth values in Figures 19 and 20 with command c = 1, where a similar conclusion is observed as for the case in the previous examples.

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Example 5 (Effects of multiple leaders on a circle graph): Finally, we again consider a circle graph topology with nonidentical number of leaders given in Figure 21. The effect of added leaders is investigated by comparing four cases. For the first case (Example 5.1), we use one leader and for the remaining three cases we, respectively use two leaders, three leaders, and four leaders (Examples 5.2–5.4, respectively). We then conducted a linear matrix inequality analysis for these four configurations. Specifically, the top subplot of Figure 22 presents the resulting linear matrix inequality feasible stability regions of allowable actuator bandwidth values and the bottom subplot of the same

figure shows the minimum unstable values obtained numerically from the simulation for a given constant command c = 1. Comparing the result of bottom subplot with the top subplot, it can be readily seen that the proposed linear matrix inequality results provide less conservative conclusions on determining the feasible stability regions. Moreover, it is clear that the change in the number of leaders significantly alters resulting feasible regions. In particular, one can conclude from Figure 22 that as number of the leaders increase, the feasible stability region increases. Finally, to show the performance of the proposed distributed adaptive control design, two points are selected from

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Figure . Circle graphs with additional links for Example  (Example . on the left subfigure, Example . in the middle subfigure, and Example . on the right subfigure) with ‘dark grey’ colour indicating the leader and ‘light grey’ colour indicating the followers.

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Figure 22 for the case of Example 5.1 which has the same graph topology as Example 4.1. Thus, the simulation results at these actuator bandwidth values are the same as in Figures 19 and 20 with command c = 1, where a similar conclusion is observed as for the case in the previous examples.

Figure . Circle graphs with a nonidentical number of leaders for Example  (Example . on the top left subfigure, Example . on the top right subfigure, Example . on the bottom left subfigure, and Example . on the bottom right subfigure) with ‘dark grey’colour indicating the leader and ‘light grey’colour indicating the followers.

6. Conclusions An open problem in distributed control design for multiagent systems is the ability of the controlled system to guarantee stability and performance with respect to often nonidentical (e.g. slow and fast) agent actuation capabilities and unknown parameters in the agent dynamics. To this end, we proposed a distributed adaptive control architecture based on a hedging method for linear time-invariant multiagent systems with heterogeneous actuator dynamics and system uncertainties. The stability of the controlled multiagent system was analysed using Lyapunov stability theory. Linear matrix inequalities were then utilised to perform stability verification in terms of numerically computing the fundamental tradeoff between heterogeneous agent actuation capabilities and unknown parameters in agent dynamics. Several

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illustrative numerical examples thoroughly show the efficacy of the proposed design procedure by investigating the effect of different leader locations in different graph topologies, the effect of additional information links, and the effect of additional leaders. As a future research direction, one can consider extending the results of this paper to the case with actuator amplitude and rate saturation, where the approaches given in, for example, Doyle, Smith, and Enns (1987), Johnson and Calise (2001), Lavretsky and Hovakimyan (2007), Leonessa, Haddad, Hayakawa, and Morel (2009), Li, Xiang, and Wei (2011), Su, Chen, Lam, and Lin (2013), Zhang, Qin, and Luo (2014), Yang, Meng, Dimarogonas, and Johansson (2014), Qin, Fu, Zheng, and Gao (2017), and Jin, Qin, Shi, and Zheng (2017) can be used to this end.

Note 1. Since the numerically obtained minimum unstable values of the pair (λ1 , λ2 ) do not significantly change in these middle and bottom subplots corresponding to time-invariant and time-varying command inputs, respectively, we only consider the case with the time-invariant command input c = 1 in the following examples.

Disclosure statement No potential conflict of interest was reported by the authors.

Funding This research was supported by the Dynamics, Control, and Systems Diagnostics Program of the National Science Foundation [grant number CMMI1561836], [grant number CMMI–1657637].

ORCID Tansel Yucelen

http://orcid.org/0000-0003-1156-1877

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