Guaranteed Cost Tracking for Uncertain Coupled Multi-agent Systems Using Consensus over a Directed GraphThis work was supported by the Australian Research Council under projects DP0987369 and DP120102152. To be presented at the 2013 Australian Control Conference, Perth, Australia

Guaranteed Cost Tracking for Uncertain Coupled Multi-agent Systems Using Consensus over a Directed Graph††thanks: This work was supported by the Australian Research Council under projects DP0987369 and DP120102152. To be presented at the 2013 Australian Control Conference, Perth, Australia

Yi Cheng School of Engineering and Information Technology, The University of New South Wales at the Australian Defence Force Academy, Canberra, ACT 2600, Australia. Email: y.cheng@adfa.edu.au, v.ougrinovski@adfa.edu.au.    V. Ugrinovskii1  The work of V. Ugrinovskii was carried in part while he was a visitor at the Australian National University.    Guanghui Wen Department of Mathematics, Southeast University, Nanjing 210096, China. Email: wenguanghui@gmail.com.
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Abstract

This paper considers the leader-follower control problem for a linear multi-agent system with directed communication topology and linear nonidentical uncertain coupling subject to integral quadratic constraints (IQCs). A consensus-type control protocol is proposed based on each agent’s states relative to its neighbors and leader’s state relative to agents which observe the leader. A sufficient condition is obtained by overbounding the cost function. Based on this sufficient condition, a computational algorithm is introduced to minimize the proposed guaranteed bound on tracking performance, which yields a suboptimal bound on the system consensus control and tracking performance. The effectiveness of the proposed method is demonstrated using a simulation example.

1 Introduction

In recent years, theoretical studies of distributed coordination and control for multi-agent systems have attracted much attention in the literature, with broad applications in various areas including unmanned air vehicles (UAVs), formation control, flocking, distributed sensor networks, etc. [1]. As a result, much progress has been made in the study of cooperative control of multi-agent systems [2, 3, 4].

Efforts have recently been made to consider the leader-following consensus problem. For example, the leader-following consensus problem for higher order multi-agent systems is presented for both fixed and switching topologies in [5]. In [6], distributed observers are designed for the system of second-order agents where an active leader to be followed moves with an unknown velocity, and the interaction topology has a switching nature. The consensus-based approach to observer-based synchronization of multi-agent systems to the leader has been explored in [7, 8].

A common feature of the above literature on leader-following consensus-based control problems is that interactions between agents are not considered. However, in many physical systems, interactions between agents are inevitable and must be taken into account. Examples of systems with a dynamical interaction between subsystems include power systems and spacecraft control systems [9]. This necessitates considering systems of interconnected agents.

In this paper, the leader-follower control problem for multi-agent systems coupled via linear unmodelled dynamics is considered. Coupling among the agents is regarded as an uncertainty and is described in term of time domain integral quadratic constraints (IQCs) [10]. The IQC modeling is a well established technique to describe uncertain interactions between subsystems in a large scale system [11, 12, 13].

The motivation of this paper is to extend our previous work [14] as follows. Firstly, this paper considers the multi-agent system with directed topology rather than undirected topology, which poses additional difficulty compared with [14], due to the Laplacian matrix of directed graphs being in general asymmetric. Therefore a different technique is used in this paper to obtain a sufficient condition for leader follower tracking which does not involve coordinate transformation; the latter was used in [14] and required the Laplacian matrix of the graph to be symmetric. Furthermore, we consider a more general, compared to [14], class of systems with nonidentical time varying uncertain coupling. In this paper, we also propose a different LQR based cost function which describes the cost on the tracking error between the leader and all of the followers. In contrast, in [14] a consensus based cost function is considered, which penalizes the system input, the state error between the agent and its neighbours, as well as the tracking error between the leader and selected agents which observe the leader. Furthermore, the graph topology of the control protocol does not need to be the same as the topology of interconnections between the agents. Even though both communication topologies are represented as directed graphs, these graphs can be different: the agents are coupled over one directed graph, but the control protocol for the system uses another directed graph.

The main contribution of this paper is to propose a sufficient condition for the design of a guaranteed performance leader-follower control protocol for multi-agent systems with directed interconnection topology and a quite general linear uncertain coupling subject to IQCs. The sufficient condition is obtained by using a direct over-bounding technique and involves checking feasibility of parameterized linear matrix inequalities (LMIs). The computational algorithm is introduced to minimize the proposed guaranteed bound by choosing local tuning parameters and guarantee a suboptimal bound on the system tracking performance.

The remainder of the paper proceeds as follows. In Section 2 of the paper, we set up the leader follower control problem for a multi-agent system with directed topology and nonidentical linear uncertain coupling and give some preliminaries. The main results are given in Section III. In section IV, the computational algorithms are introduced. Section V gives an example which illustrates the theory presented in the paper. Finally, the conclusions are given in Section VI.

2 Problem Formulation and Preliminaries

2.1 Graph theory

Consider a directed graph , where is a finite nonempty node set and is an edge set of ordered pairs of nodes. The edge in the edge set of an directed graph means that the node can obtain information from node . Node is called a neighbor of node if . The set of neighbors of node is defined as . is a simple graph if it has no self-loops or repeated edges. If there is a directed path between any two nodes of the graph , then the graph is strongly connected. The adjacency matrix of the directed graph is defined as if , and otherwise. The in-degree matrix is a diagonal matrix, whose diagonal elements are for . Also, let be the out-degree of node . The Laplacian matrix of the graph is defined as

2.2 Problem Formulation

Consider a system consisting of agents and a leader. All agents are assumed to be linear dynamical agents, coupled with their neighbors via, in general nonidentical, linear uncertain coupling. The connection between agents is described by a directed graph , with the node set , an edge set and a corresponding adjacency matrix . The dynamics of the th agent are described as

 ˙xi=Axi+B1ui+B2∑j∈Siφij(t,xj(.)|t0−xi(.)|t0), (1)

where the summation is over the set of neighbors of node in the graph . The notation describes a linear uncertain operator mapping functions , into . Also, is the state, is the control input. We note that the last term in (1) reflects a relative, time varying nature of interactions between agents.

Let be the space of functions such that .

Assumption 1

All the mappings satisfy the following assumptions:

1. .

2. is linear in the second argument, i.e., if , then .

3. Given a matrix , there exists a sequence , such that for every , the following IQC holds

 ∫tl0∥φij(t,y(.)|t0)∥2dt≤∫tl0∥Cijy∥2dt, (2) ∀y∈L2e[0,∞).

The sequence is assumed to be the same for all . The class of operators that satisfy these assumptions will be denoted by . We note that matrices are assumed to be fixed.

In addition to the system (1), suppose a leader is given. The dynamics of the leader, labeled , is expressed as

 ˙x0=Ax0, (3)

where is its state. The control communication topology between agents is described by a directed graph , with the same node set , but possibly different edge set and a corresponding adjacency matrix . The Laplacian matrix of the graph is denoted as . We assume throughout the paper that the leader node can be observed from a subset of nodes of the graph . If the leader is observed by the node , we extend the graph by adding the edge with weighting gain , otherwise let . We refer to node with as a pinned or controlled node. The diagonal matrix is commonly referred to as the pinning matrix. The system is assumed to have at least one agent connected to the leader, hence .

Define error vectors as , . Then dynamics of the synchronization errors satisfy the equation

 ˙ei=Aei−B1ui−B2∑j∈Siφij(t,ei(.)|t0−ej(.)|t0). (4)

In this paper we are concerned with finding a control protocol for each node of the form

 ui=−K{∑j∈Ti(xj−xi)+gi(x0−xi)}, (5)

where is the feedback gain matrix to be found, and is the set of neighbors of node in the graph . As a measure of system performance, we will use the quadratic cost function,

 J(u) =N∑i=1∫∞0(e′iQei+u′iRui)dt, (6)

where and are given weighting matrices.

Remark 1

In [14] we considered a different cost function,

 J′(u)=N∑i=1∫∞0(12∑j∈Ni(ei−ej)′Q(ei−ej) +gie′iQei+u′iRui)dt.

Each addend in this cost function penalizes the th system input, the disagreement between the th and the th system states, where is a neighbor of , as well as the tracking errors between the leader and the pinned agents which observe the leader. In contrast, the cost function (6) in this paper describes the cost on the tracking error between the leader and all of the followers and system input.

Taking linearity of the operator into account, the synchronization error dynamic (4) can be represented as

 ˙ei=Aei−B1ui−B2∑j∈Si(φij(t,ei(.)|t0)−φij(t,ej(.)|t0)). (7)

The problem in this paper is to find a control protocol (5) which solves the leader following consensus control problem as follows:

Problem 1

Under Assumption 1, find a control protocol of the form (5) such that

 supΞ0J(u)<∞. (8)

Here means that the supremum is taken over the set of all operators that belong to the class of operators. Since , then (8) implies

 ∫∞0∥ei∥2dt<∞∀i=1,…,N. (9)

Hence, solving Problem 1 implies synchronization of all agents to the leader in the sense.

3 The Main Result

In this section, the main result of this paper is presented which is a sufficient condition for the system (1) to be able to track the leader with guaranteed tracking performance.

First we present the following result of [15] and some notation.

Assumption 2

The digraph contains a spanning tree and the root node obtains information from the leader node, i.e., .

Lemma 1

([15]) Under Assumption 2, is nonsingular. Define , and , then and .

Let be the maximum eigenvalue of and . Also let . According to Lemma 1, is a positive definite and symmetric matrix. Let be an orthogonal matrix such that

 T−1MT=J=diag[λ1,⋯,λN], (10)

and denote . For node of the graph , introduce matrices , , where are the elements of the neighbourhood set , and are the nodes with the property ; and are the in-degree and the out-degree of node , respectively, in the graph . Also, introduce the matrix .

Theorem 1

Let a matrix , and constants , , , , exist such that the following LMIs are satisfied simultaneously

 ⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣ZiYQ1/2Y^C′iY¯C′iQ1/2Y−1ϑiI00^CiY0−Φi0¯CiY00−Ωi⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦<0, (11)

where

 Zi= AY+YA′−σϑiB1^R−1B′1 +ϑi∑j∈Si(1νij+1μij)B2B′2, Φi= diag[ϑiνij,j∈Si], Ωi= diag[ϑiμji,j:i∈Sj].

Then the control protocol (5) with solves Problem 1. Furthermore, this protocol guarantees the following performance bound

 supΞ0J(u)≤N∑i=1ϑ−1ie′i(0)Y−1ei(0). (12)

Proof: Using the Schur complement, the LMIs (11) can be transformed into the following Riccati inequality

 AY+YA′−σϑiB1^R−1B′1+ϑi∑j∈Si(1νij+1μij)B2B′2 +Y(ϑiQ+ϑ−1i(∑j∈SiνijC′ijCij +∑j:i∈SjμjiC′jiCji))Y<0. (13)

After pre- and post-multiplying (3) by and multiplying (3) by , then substituting into the Riccati inequality (3), we obtain

 Y−1(ϑ−1iA+σB1K)+(ϑ−1iA+σB1K)′Y−1 +σK′^RK+∑j∈Si(1νij+1μij)Y−1B2B′2Y−1+Q +ϑ−2i(∑j∈SiνijC′ijCij+∑j:i∈SjμjiC′jiCji)<0. (14)

Define and consider the following Lyapunov function candidate for the subsystems (7):

 V(e)=N∑i=1ϑ−1ie′iY−1ei. (15)

Then

 dV(e)dt =N∑i=12e′iY−1(ϑ−1iAei +ϑ−1iB1K(∑j∈Ti(ei−ej)+giei)) −2N∑i=1ϑ−1i∑j∈Sie′iY−1B2φij(t,ei(.)|t0) +2N∑i=1ϑ−1i∑j∈Sie′iY−1B2φij(t,ej(.)|t0). (16)

Note the following inequality:

 N∑i=12e′iϑ−1iY−1B1K(∑j∈Ti(ei−ej)+giei) =2e′(Θ(L2+G)⊗(Y−1B1K))e =e′((Θ(L2+G)+(L2+G)′Θ)⊗(Y−1B1^R−1B′1Y−1))e =y′((Θ(L2+G)+(L2+G)′Θ)⊗Ip)y ≤2σy′(IN⊗Ip)y=2σe′(IN⊗Y−1B1^R−1B′1Y−1)e =2N∑i=1σe′iY−1B1^R−1B′1Y−1ei, (17)

where .

From (3) and (3), one has

 dV(e)dt ≤N∑i=12e′iY−1(ϑ−1iA+σB1K)ei −2N∑i=1ϑ−1i∑j∈Sie′iY−1B2φij(t,ei(.)|t0) +2N∑i=1ϑ−1i∑j∈Sie′iY−1B2φij(t,ej(.)|t0). (18)

Substituting the Riccati inequality (3) into (3), we have

 dV(e)dt ≤−N∑i=1e′i(σK′^RK+Q +∑j∈Si(1νij+1μij)Y−1B2B′2Y−1 +ϑ−2i(∑j∈SiνijC′ijCij+∑j:i∈SjμjiC′jiCji))ei −2N∑i=1ϑ−1i∑j∈Sie′iY−1B2φij(t,ei(.)|t0) +2N∑i=1ϑ−1i∑j∈Sie′iY−1B2φij(t,ej(.)|t0). (19)

Using the following identity,

 N∑i=1∑j∈Siμije′jC′ijCijej=N∑i=1∑j:i∈Sjμjie′iC′jiCjiei,

one has

 dV(e)dt≤−N∑i=1e′i(σK′^RK+Q)ei −N∑i=1∑j∈Si∥1√νijB′2Y−1ei+√νijϑ−1iφij(t,ei(.)|t0)∥2 +N∑i=1∑j∈Siϑ−2iνij(∥φij(t,ei(.)|t0)∥2−∥Cijei∥2) −N∑i=1∑j∈Si∥1√μijB′2Y−1ei−√μijϑ−1iφij(t,ej(.)|t0)∥2 +N∑i=1∑j∈Siϑ−2iμij(∥φij(t,ej(.)|t0)∥2−∥Cijej∥2). (20)

According to the IQC condition (2), we have

 ∫tl0dV(e)dtdt≤−N∑i=1∫tl0e′i(σK′^RK+Q)eidt. (21)

Since , then (21) implies

 N∑i=1∫tl0e′i(σK′^RK+Q)eidt≤V(e(0)). (22)

The expression on the right hand side of the above inequality is independent of . Letting leads to

 N∑i=1∫∞0e′i(σK′^RK+Q)eidt≤V(e(0)). (23)

Using (6) and (5), we have

 J(u) =N∑i=1∫∞0(e′iQei+u′iRui)dt =∫∞0(e′(IN⊗Q)e +e′[(L2+G)′(L2+G)⊗K′RK]e)dt ≤∫∞0(e′(IN⊗Q)e+e′[IN⊗¯λK′RK]e)dt =N∑i=1∫∞0e′i(¯λK′RK+Q)eidt. (24)

Since , then we obtain

 J(u) ≤N∑i=1∫∞0e′i(σK′^RK+Q)eidt ≤N∑i=1ϑ−1ie′i(0)Y−1ei(0). (25)

It implies that the control protocol (5) with solves Problem 1, and also guarantees the performance bound (12).

4 The Computational Algorithm

In this section, we provide an algorithm to calculate a suboptimal control gain . According to Theorem 1, the upper bound on tracking performance is given by the right hand side of (12). Hence, one can achieve a suboptimal guaranteed performance by optimizing this upper bound over the feasibility set of the LMIs (11):

 J∗(???)=infN∑i=1ϑ−1ie′i(0)Y−1ei(0), (26)

where the infimum is taken over the feasibility set of the LMIs (11), .

As in [12], the optimization problem (26) can be shown to be equivalent to the minimization of subject to the constraints

 γ>N∑i=1ϑ−1ie′i(0)Y−1ei(0), i=1,⋯,N. (27)

By the Schur complement, (27) is equivalent to the LMI

 [γe′(0)e(0)Υ]>0, i=1,⋯,N, (28)

where

 e(0)= [e1(0)′ e2(0)′ … eN(0)′]′, Υ= diag[ϑiY,i=1,2,⋯,N].

This leads us to introduce the following optimization problem in the variables and : Find

 J∗(???),(???)≜infγ, (29)

where the infimum is with respect to and subject to (11) and (28).

We conclude this discussion by stating equivalence between the optimization problems (26) and (29).

Theorem 2

.

Proof: The proof of this theorem is similar to the proof of Theorem 15 in [12].

Based on the foregoing discussion, we propose an algorithm for the design of the suboptimal protocol (5) based on Theorems 1 and 2:

• Solve the optimization problem (29), to a desired accuracy, obtaining a feasible collection , , and . The collection then belongs to the feasibility set of the LMIs (11).

• Using the found matrix , construct the gain matrix to be used in (5), by letting . Also, the guaranteed consensus performance bound for this protocol can be computed, using the expression on the right-hand side of equation (12).

It must be noted that the above algorithms require the knowledge of initial conditions of the leader and agents. In practice, however, the initial state of the leader may not be known. It is possible to avoid using in these algorithms using the approaches outlined in [12].

5 Example

To illustrate the proposed method, consider a system consisting of three identical pendulums coupled by two spring-damper systems. Each pendulum is subject to an input as shown in Fig. 1. The dynamics of the coupled system are governed by the following equations

 ml2¨α1= −k11a2(t)(α1−α2)−k12a2(t)(˙α1−˙α2) −mglα1−u1, ml2¨α2= −k21b2(t)(α2−α3)−k11a2(t)(α2−α1) −k22b2(t)(˙α2−˙α3)−k12a2(t)(˙α2−˙α1) (30) −mglα2−u2, ml2¨α3= −k21b2(t)(α3−α2)−k22b2(t)(˙α3−˙α2) −mglα3−u3,

where is the length of the pendulum, and are the positions of the spring-damper, is the gravitational acceleration constant, is the mass of each pendulum, and are the spring constant and damping coefficient for the leftmost spring-damper pair, while and are the spring constant and damping coefficient for the rightmost spring-damper pair. The position of the spring-damper can change along the full length of the pendulums and is considered to be uncertain, that is , .

In addition to the three pendulums, consider the leader pendulum which is identical to those given. Its dynamics are described by the equation

 ml2¨α0=−mglα0. (31)

Choosing the state vectors as , , and , equations (5) and (31) can be written in the form of (1), (3) , where , , and , , , ,

The agents in this example are coupled according to the undirected graph shown in Fig. 2, which is treated here as a special case of directed graph with symmetric adjacency matrix. On the other hand, the control communication topology of the system is assumed to be a linear directed graph shown in Fig. 3. According to this graph, only agent observes the leader. The Laplacian matrix of the graph consisting of nodes 1, 2 and 3 and the pinning matrix are

 L2=⎡⎢⎣000−1100−11⎤⎥⎦,  G=⎡⎢⎣100000000⎤⎥⎦.

To illustrate the design based on Theorems 1 and 2, the LMI problem in Theorem 2 was solved numerically, and then the trajectories of the coupled pendulum system with the obtained protocol were simulated. To this end, the parameters of the coupled pendulum system were chosen to be , , , , , , , , . In the cost function, we let and . Using the computational algorithm based on Theorem 2, the problem (26) was found to be feasible and yielded the gain matrix . The performance bound was minimized by , with parameters , , , , , , , . The simulation results for this protocol are shown in Fig. 4. Also, using the controller obtained by means of the computational algorithm proposed in Theorem 2, we directly computed the performance cost (6) for the system to be , while the theoretically predicted bound is .

6 Conclusions

The consensus control for leader-tracking problem with guaranteed tracking performance for nonidentical uncertain coupled linear systems connected over a directed graph has been discussed in this paper. A sufficient condition was proposed by using the direct overbounding of the performance cost. According to the simulation results, the proposed computational algorithm based on Theorems 1, 2, which solve N coupled LMIs, guarantees a suboptimal performance.

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