2013 IEEE Congress on Evolutionary Computation June 20-23, Cancún, México
Binary Multiagent Coordination Optimization with Application to Formation Control Design Haopeng Zhang and Qing Hui Department of Mechanical Engineering Texas Tech University, Lubbock, Texas 79409–1021, USA Email:
[email protected];
[email protected]
Abstract—In this paper, a novel binary swarm optimization algorithm, called Binary Multiagent Coordination Optimization (BMCO) algorithm, is proposed by introducing a communication topology for the particles in the algorithm and using recently developed multiagent consensus protocols from control theory. Due to the consensus term embedded into the update formula for the velocity, the BMCO algorithm shows a faster convergence rate than the standard Binary Particle Swarm Optimization (BPSO). We use eight benchmark functions to test the performance of the standard BPSO, BMCO, and a variation of BPSO called Novel BPSO (NBPSO). The optimal values and convergence rates of these three algorithms are provided and compared. From the numerical results, we can conclude that the performance of the BMCO algorithm is superior to that of BPSO and NBPSO. Next, as an application, we use the proposed algorithm to solve a topology optimization problem for an observer-based multiagent formation control design. In the existing literature, the topologies for positions and velocities of multiple agents are always assumed to be the same. However, in our proposed formation control protocol, these two topologies are not necessarily the same, not even all connected. To this extent, designing optimal heterogeneous topologies for the formation control protocol under the tradeoff between communication cost and operation time can be formulated as a binary optimization problem. Finally, A numerical illustration is provided by comparing the three binary algorithms to solve the topology optimization problem for multiagent formation control, and the BMCO algorithm shows the best result compared with BPSO and NBPSO. Index Terms—Binary optimization, multiagent coordination, particle swarm optimization, swarm intelligence, formation control.
I. I NTRODUCTION It is well known that a wide variety of nonlinear and combinatorial problems can be formulated as a binary optimization problem, wherein the variable can only take 0 or 1, for instance, the extensively studied Traveling-Salesman Problem (TSP) [1]. Similar to TSP, the optimization of distributed network systems underlying a communication topology is another application of the binary optimization [2], [3]. In this problem, if agent 𝑖 communicates with agent 𝑗, then we set the linkage variable 𝑥𝑖,𝑗 to be 1, otherwise 𝑥𝑖,𝑗 = 0. In [2], this type of the topology design optimization problem is proposed in which the objective function is the tradeoff between the convergence rate to consensus and communication link cost. This work was supported by the Defense Threat Reduction Agency, Basic Research Award #HDTRA1-10-1-0090, to Texas Tech University.
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To efficiently solve such binary optimization problems, the Binary Particle Swarm Optimization (BPSO) algorithm is proposed in [4], where the trajectories of the particles are changes in the probability so that the coordinates of the particles will take on a zero or one value. With the aim of making the BPSO algorithm more applicable to a larger class of binary optimization problems, some variations of the BPSO algorithm are developed. In particular, by introducing birth and death operations in order to make the population very dynamic, a Genetic BPSO (GBPSO) model is proposed in [5]. Compared to the original Particle Swarm Optimization (PSO) model, this new strategy proposes a more natural simulation of the social behavior of intelligent animals, and the experimental results reveal that the GBPSO model can reach broader domains in the search space and converge faster in very high dimensional and complex environments. In stead of extending the search space, [6] employs a trigonometric function as a bit string, by which the larger dimensional binary space can be represented by a smaller 4-dimensional continuous space. A Novel BPSO (NBPSO) is proposed in [7] where the velocity vector of the original BPSO is redefined, leading to a better interpretation of turning PSO into BPSO. In this paper, a new Binary Multiagent Coordination Optimization (BMCO) algorithm is proposed by introducing the communication topology and cooperative control protocols to the particle’s velocity and position information in a BPSO-like algorithm during the searching process. Cooperative control for multiagent system has been extensively studied in the literature [8]–[10], where the topology for the multiagent system plays a significant role to fulfil the control protocols. In cooperative coordination point of view, the particles in the standard BPSO work cooperatively to approach the optimal solution of a particular problem without intentionally communicating with other particles. Some research has been conducted by sharing the local optimal solution between neighbors underlying a communication topology [11], [12] for the PSO algorithm. Instead of sharing local optimal solutions, the BMCO algorithm updates the particle’s velocity by two “consensus” terms of the previous velocity and position. Through the test of several benchmark functions, the BMCO shows faster convergence and more accurate optimal solutions as opposed to the BPSO and NBPSO algorithms. Formation control for multiagent systems has gained consid-
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erable attention in control theory due to its wide applications in military and civil aspects [13]–[16]. However, in most double-integrator based multiagent systems, the communication topologies for the velocity and position of agents are supposed to be the same and connected in the most literature [16], [17]. In practical scenarios, the sensors for detecting the information of velocity may be different from that of position. Another promising advantage of studying different topologies for position and velocity is the extensive choice of topologies. In cost control point of view, the communication expense depends on the communication topologies, therefore the controlling cost may be lower if we choose the communication topologies with lower communication expenditures. As a binary optimization application, in this paper, we study a particular formation control protocol [18] for multiagent systems underlying three communication topologies, and develop a weaker condition for the connectivity of the communication topologies to achieve the desired formation than the available condition in [18]. Next, by considering the communication cost and operation time for multiagent formation control, a heterogeneous topology optimization problem is formulated. Moreover, the BPSO, BMCO, and NBPSO algorithms are applied to solve this optimization problem and compared with each other in terms of numerical performance and convergence. The outline of the paper is as follows: Mathematical preliminaries are introduced in Section II, such as the definition of graph topology and the algorithmic description of BPSO. The BMCO algorithm is proposed in Section III. The performance of BMCO is provided in Section IV, in which the optimal value and convergence rate are presented respectively by solving eight benchmark functions for BPSO, BMCO and NBPSO. In Section V, a heterogeneous topology optimization problem is formulated for formation control design. This optimization problem is solved numerically by the BPSO, BMCO and NBPSO algorithms in Section VI. Finally, Section VII concludes the paper.
II. M ATHEMATICAL P RELIMINARIES A. Graph Topology In this paper, undirected or directed graphs are used to represent a topology of communication networks. Specifically, let 𝒢 = (𝒱, ℰ) be a directed graph (or digraph) denoting the communication network with the set of nodes (or vertices) 𝒱 = {1, . . . , 𝑞} involving a finite nonempty set denoting the agents, and the set of edges ℰ ⊆ 𝒱 × 𝒱 involving a set of ordered pairs (𝑖, 𝑗) denoting the direction of communication. A graph or undirected graph 𝒢 is a directed graph for which the arc set is symmetric. For graphs, we use unordered pairs {𝑖, 𝑗} for edges. The set of neighbors of node 𝑖 is thus defined by 𝒩𝑖 = {𝑗 ∈ 𝒱 : {𝑖, 𝑗} ∈ ℰ}. The connectivity matrix 𝐿 associated with the graph 𝒢 is
defined by 𝐿𝒢𝑖,𝑗
𝐿𝒢𝑖,𝑖
{
0, if (𝑖, 𝑗) ∕∈ ℰ, 1, otherwise, 𝑖 ∕= 𝑗, 𝑖, 𝑗 = 1, . . . , 𝑞, 𝑞 ∑ − 𝐿𝑖,𝑘 , 𝑖 = 1, . . . , 𝑞,
≜
≜
(1) (2)
𝑘=1, 𝑘∕=𝑖
where 𝑞 is the number of agents. If there is a path of edges from any node to any other node in the graph, then we call the graph connected. B. BPSO Algorithm The standard BPSO algorithm is developed in [4]. This algorithm is motivated by PSO and optimizes a problem by having a population of candidate solutions, here dubbed particles, and moving these particles around in the search-space according to the following simple mathematical formulae over the particle’s position and velocity: 𝑣𝑖 (𝑘 + 1)
=
𝑥𝑖,𝑗 (𝑘 + 1)
=
𝜇𝑣𝑖 (𝑘) + 𝜆(𝑝𝑖 (𝑘) − 𝑥𝑖 (𝑘)) +𝜅(𝑝 − 𝑥𝑖 (𝑘)); { 1 if 𝑟𝑖,𝑗 < sig(𝑣𝑖,𝑗 (𝑘 + 1)) 0 otherwise
(3)
where, 𝑥𝑖 (𝑘) and 𝑣𝑖 (𝑘) are the position and velocity vector for particle 𝑖 at step 𝑘 respectively, and 𝑝𝑖 (𝑘), 𝑝 are the local optimal solution at step 𝑘 and global optimal solution respectively, 𝑥𝑖,𝑗 (𝑘 + 1), 𝑣𝑖,𝑗 (𝑘 + 1) are the 𝑗th component of 𝑥𝑖 (𝑘 + 1) and 𝑣𝑖 (𝑘 + 1) respectively, 𝑟𝑖,𝑗 , 𝜂, 𝜇, 𝜅 are random numbers in [0,1], and sig function is defined by 1 (4) 1 + 𝑒−𝑥 III. B INARY M ULTIAGENT C OORDINATION O PTIMIZATION A LGORITHM sig(𝑥) =
The BMCO algorithm, which is shown in Algorithm 1, is a new optimization technique based not only on the natureinspired social behavior metaphor, but also on cooperative control of multiple agents. Similar to BPSO, the BMCO algorithm consists of a number of particles called agents moving in the solution space to minimize 𝐽(𝑥). It starts with a set of random solutions for agents that can communicate with each other in a certain topology. The agents then move through the solution space based on the evaluation of their cost and neighbor-to-neighbor rules inspired by multiagent consensus protocols [8]–[10], [19]. As the algorithm progresses, the agents will accelerate towards individuals with better cost values. IV. BMCO A LGORITHM P ERFORMANCE A. Test Function Review
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∙
∑𝑛 2 Sphere function: 𝑓 (𝑥) = 𝑖=1 𝑥𝑖 . The test area is usually restricted to the hypercube −5.12 ≤ 𝑥𝑖 ≤ 5.12 or −30 ≤ 𝑥𝑖 ≤ 30, 𝑖 = 1, . . . , 𝑛. The global minimum of 𝑓 (𝑥) is 0 at 𝑥𝑖 = 0.
Algorithm 1 Binary Multiagent Coordination Optimization Algorithm for each agent 𝑖 = 1, . . . , 𝑞 do Initialize the agent’s position with a uniformly distributed random vector: 𝑥𝑖 ∼ 𝑈 (𝑥, 𝑥), where 𝑥 and 𝑥 are the lower and upper boundaries of the search space; Initialize the agent’s velocity: 𝑣𝑖 ∼ 𝑈 (𝑣, 𝑣), where 𝑣 and 𝑣 are the lower and upper boundaries of the search speed; Update the agent’s best known position to its initial position: 𝑝𝑖 ← 𝑥𝑖 ; If 𝐽(𝑝𝑖 ) < 𝐽(𝑝) update the multiagent network’s best known position: 𝑝 ← 𝑝𝑖 . end for repeat 𝑘 ← 𝑘 + 1; for each agent 𝑗 = 1, . . . , 𝑞 do Choose random parameters: 𝜂, 𝜇, 𝜆, 𝜅 ∼ 𝑈 (0, Θ), Θ > 0; Update the agent’s velocity: ∑ ∑ (𝑣𝑗 −𝑣𝑖 )+𝜆 (𝑥𝑗 −𝑥𝑖 )+𝜅(𝑝−𝑥𝑖 ); 𝑣𝑖 ← 𝜇𝑣𝑖 +𝜂 𝑗∈𝒩𝑖
𝑗∈𝒩𝑖
Update the agent’s position: { 1 if 𝑟𝑖,𝑗 < sig(𝑣𝑖,𝑗 ) 𝑥𝑖,𝑗 ← 0 otherwise
∙
∙
∙
13
f(k)
12
11
10
9
8
7
0
10
Fig. 1.
∙
∙
∙
(6)
∑𝑛 2 Rastrigin function: 𝑓 (𝑥) = 10𝑛 + 𝑖=1 [𝑥𝑖 − 10 cos(2𝜋𝑥𝑖 )]. The test area is usually restricted to the hypercube −30 ≤ 𝑥𝑖 ≤ 30, 𝑖 = 1, . . . , 𝑛. The global minimum of 𝑓 (𝑥) is 0 at 𝑥𝑖 = 0. De Jone’s f2: 𝑓 (𝑥) = 100(𝑥21 − 𝑥22 )2 + (1 − 𝑥1 )2 . The test area is usually restricted to the hypercube −2.408 ≤ 𝑥𝑖 ≤ 2.408, 𝑖 = 1, 2. The global minimum of 𝑓 (𝑥) is 0 at 𝑥𝑖 = 1. √ (sin 𝑥21 +𝑥22 )2 −0.5 Schwefel’s f6 function: 𝑓 (𝑥) = 0.5 + 1+0.001(𝑥 2 +𝑥2 )2 . 1 2 The test area is usually restricted to the hypercube −100 ≤ 𝑥𝑖 ≤ 100, 𝑖 = 1, 2. The global minimum 𝑓 (𝑥) is 0. ∑𝑛 1 2 Griewank’s function: 𝑓 (𝑥) = 𝑖=1 𝑥𝑖 − 4000 ∏𝑛 𝑥 √𝑖 𝑖=1 cos( 𝑖 ) + 1. The test area is usually restricted to the hypercube −600 ≤ 𝑥𝑖 ≤ 600, 𝑖 = 1, . . . , 𝑛. The global minimum of 𝑓 (𝑥) is 0 at 𝑥𝑖 = 0.
BMCO BPSO NBPSO
14
(5)
where 𝑟𝑖,𝑗 ∼ 𝑈 (0, 1); 𝑥𝑖 ← [𝑥𝑖,1 , . . . , 𝑥𝑖,𝑞 ]T ; for 𝐽(𝑥𝑖 ) < 𝐽(𝑝𝑖 ) do Update the agent’s best known position: 𝑝𝑖 ← 𝑥𝑖 ; If 𝐽(𝑝𝑖 ) < 𝐽(𝑝) update the multiagent network’s best known position: 𝑝 ← 𝑝𝑖 ; end for end for until 𝑘 is large enough or the change of 𝑓 becomes small return 𝑝
∙
15
20
30
40 50 60 Iteration index
70
80
90
100
Test function: Sphere function
∑20 2 2 Zakharov function: 𝑓 (𝑥) = 𝑖=1 𝑥𝑖 + (0.5𝑖𝑥𝑖 ) + 4 (0.5𝑖𝑥𝑖 ) . The test area is usually restricted to the hypercube −10 ≤ 𝑥𝑖 ≤ 10, 𝑖 = 1, . . . , 20. The global minimum of 𝑓 (𝑥) is 0 at 𝑥𝑖 = 0. Levy function: ∑ 𝑓 (𝑥) = sin2 (𝜋𝑥1 ) + (𝑥𝑛 − 1)2 (1 + 𝑛−1 2 sin (2𝜋𝑥𝑛 )) − 𝑖=1 (𝑥𝑖 − 1)2 (1 + 10 sin2 (𝜋𝑥𝑖 + 1)). The test area is usually restricted to the hypercube −10 ≤ 𝑥𝑖 ≤ 10, 𝑖 = 1, . . . , 𝑛. The global minimum of 𝑓 (𝑥) is 0 at 𝑥𝑖 = 1. Colville function: 𝑓 (𝑥) = 100(𝑥1 − 𝑥21 )2 + (1 − 𝑥1 )2 + 90(𝑥4 − 𝑥23 )2 + (1 − 𝑥3 )2 + 10.1((𝑥2 − 1)2 + (𝑥4 − 1)2 ) + 19.8(𝑥2 − 1)(𝑥4 − 1). The test area is usually restricted to the hypercube −10 ≤ 𝑥𝑖 ≤ 10, 𝑖 = 1, . . . , 4, and the global minimum of 𝑓 (𝑥) is 0 at 𝑥𝑖 = 1.
B. Statistic Analysis In this subsection, statistic results of the BPSO, BMCO and NBPSO algorithms under these test functions will be presented. The search areas and some dimensions of objective functions are listed in Subsection IV-A, where 𝑛 = 30. By running 20 times for every binary optimization algorithm, the maximum, minimum, average, and median objective values are compared between the standard BPSO, NBPSO and BMCO algorithms, which are shown in Table I. From Table I, one can conclude that the proposed BMCO algorithm is better than the BPSO and NBPSO algorithms in most cases. C. Convergence Comparison In this subsection, the convergence rates of BPSO, BMCO, and NBPSO are compared by solving the relevant optimization problems for the eight test functions, which are shown in Fig. 1–8. In Fig. 4–5 and Fig. 8, the convergence rate for BMCO is faster than that of the BPSO and NBPSO algorithms, moreover, in Fig. 1, Fig. 2 and Fig. 5–7, the BMCO algorithm achieves the best optimal values among BPSO and NBPSO while in Fig. 3 and Fig. 8 both BMCO and NBPSO achieves the optimal value 0 and in Fig. 4 all three algorithms obtains the optimal value 0. Therefore, BMCO algorithm achieves the fastest convergence rate in most cases while obtains the best optimal values. V. A PPLICATION TO M ULTIAGENT F ORMATION Formation control for multiagent systems has been extensively studied due to its wide applications in military and civil
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TABLE I N UMERICAL C OMPARISON B ETWEEN BPSO, BMCO, AND NBPSO Function
Min BMCO NBPSO 0 0 3 0 0 0 0 0 0 0 0 0 0 0 7.5E-4 0 0 1.3 1.50E-32 1.50E-32 1.50E-32 0 0 0
Max BPSO BMCO NBPSO 1.6E1 1.4E1 1.5E1 2.9E1 7.1E2 1.0E3 1 1 1.0E2 4.5E-1 3.2E-1 4.5E-1 4.0E-3 4.0E-3 4.0E-3 1.6E+5 1.5E+5 1.4E+5 6.70E-1 6.36E-1 8.18E-1 4.2E1 4.2E1 1.1E2
BPSO
Sphere Rastrigin De Jong’s f2 Schaffer’s f6 Griewank Zakharov Levy Colville
Median BPSO BMCO NBPSO 1.0E1 0 1.2E1 0 0 7.1E2 0 0 0 0 0 4.1E-2 0 0 2.5E-3 0 0 9.7E+4 1.50E-32 1.50E-32 5.45E-1 0 0 0
Average BPSO BMCO NBPSO 8.5 4.05 1.2E2 1.2E1 5.7E1 7.0E2 1.5E-1 5.0E-2 2.0E1 1.2E-1 3.2E-2 1.4E-1 1.0E-3 5.0E-4 2.7E-3 5.5E+4 1.4E+4 9.6E+4 1.99E-1 8.18E-2 5.52E-1 1.1E1 6.4 1.6E1
0.9
3400 BMCO BPSO NBPSO
3200
BMCO BPSO NBPSO
0.8
3000
0.7
2800 0.6 2600
f(x)
0.5 2400
0.4 2200 0.3 2000 0.2
1800
0.1
1600 1400
0
10
Fig. 2.
20
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40 50 Iteration index
60
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0
80
Test function: Rastrigin function
0
5
Fig. 5.
10
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20 25 Iteration index
30
35
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Test function: Griewank’s function: 4
12
100 BMCO BPSO NBPSO
80
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BMCO BPSO NBPSO
7
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4 3
10 0
x 10
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f(x)
f(k)
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Fig. 3.
1
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5 6 Iteration index
7
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Fig. 6.
Test function: De Jone’s f2 function
network systems. However, in most literature on multiagent control, the communication topologies for the velocity and position are always assumed to be the same and connected. In practical scenarios, the sensors for detecting the information of velocity may be different from that of position. Another potential advantage of studying different topologies for the position and velocity is the variety of choices on topologies.
10 15 Iteration index
20
25
Test function: Zakharov function
In cost effectiveness point of view, the communication expense depends on the communication topologies, therefore the controlling cost may be lower if we choose the communication topologies with lower communication expenses. In this section, we study a particular formation control protocol [18] for multiagent systems underlying three communication topologies, and develop a weaker condition for
2.2
1 BMCO BPSO NBPSO
0.9
BMCO BPSO NBPSO
2
0.8
1.8 0.7
1.6
f(x)
f(x)
0.6 0.5 0.4
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0.1 0
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Fig. 4.
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Test function: Schaffer’s f6 function
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Fig. 7.
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Test function: Levy function
200 BMCO BPSO NBPSO
180 160 140
f(x)
120 100 80 60 40 20 0
0
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4
Fig. 8.
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10 12 Iteration index
14
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20
Test function: Colville function
the connectivity of the communication topologies to achieve the desired formation than the result in [18]. The control system we consider here is shown in (7), where 𝑥𝑖 is the position state for agent 𝑖 and 𝑥𝑐𝑖 is the observer’s position for agent 𝑖. Moreover, 𝑤𝑖𝑗 is the formation control variable which defines the distance between 𝑥𝑖 and 𝑥𝑗 . At this context, one dimensional formation control is studied. 𝑥˙ 𝑐𝑖 (𝑡)
=
− −
𝑞 ∑ 𝑗=1,𝑗∕=𝑖 𝑞 ∑
𝐿𝒢𝑖,𝑗2 (𝑥𝑐𝑖 (𝑡) − 𝑥𝑐𝑗 (𝑡)) 𝐿𝒢𝑖,𝑗3 (𝑥𝑖 (𝑡) − 𝑥𝑗 (𝑡) − 𝑤𝑖,𝑗 )
𝑗=1,𝑗∕=𝑖
𝑥˙ 𝑖 (𝑡)
=
𝑞 ∑
𝐿𝒢𝑖,𝑗1 (𝑥𝑐𝑖 (𝑡) − 𝑥𝑐𝑗 (𝑡))
(7)
𝑗=1,𝑗∕=𝑖
The equivalent matrix form of this control system can be written as ˙ (8) 𝑋(𝑡) = Φ𝑋(𝑡) + Ψ, 𝑋(0) = 𝑋0 ] [ −𝐿2 −𝐿3 and 0 denotes a zero matrix or where Φ = 𝐿1 0 2 vector. Define 𝑃 (𝑠) = 𝑠 𝐿1 + 𝑠𝐿2 + 𝐿3 , where 𝑠 is a complex variable. The following lemma in [18] gives an equivalent vector form of Ψ. [ Lemma ] 5.1: Given (8), the matrix Ψ can be represented as [ ]T −1 0 ⊗ 𝐿3 × 0 𝑤12 ⋅ ⋅ ⋅ 𝑤1𝑞 0 ⋅ ⋅ ⋅ 0 , where 0 0 ⊗ denotes the Kronecker product. Next, recall the following result from [18] for all three connected topologies case. Theorem 5.1: Consider (8) with connected graphs 𝒢𝑖 , 𝑖 = 1, 2, 3, such that 𝐿1 = 𝐿2 = 𝐿3 . Then the state of each agent in (8) achieves the desired formation given by [ T ] 11 0 𝑋0 lim𝑡→∞ 𝑋(𝑡) = 1𝑞 0 11T ]T [ 0 ⋅ ⋅ ⋅ 0 0 𝑤12 ⋅ ⋅ ⋅ 𝑤1𝑞 + [ ] ∑𝑞 0 − 1𝑞 𝑗=2 𝑤1,𝑗 ⊗ , (9) 1 lim𝑡→∞ (𝑥𝑖 (𝑡) − 𝑥𝑗 (𝑡)) = 𝑤𝑖,𝑗 (10)
In this section, we consider the case where 𝒢1 and 𝒢3 are connected but not necessarily the same topology, and 𝒢2 is disconnected but not zero. We have the following result regarding the convergence of (8). Theorem 5.2: Consider (8). Assume that 𝒢1 and 𝒢3 are connected but not necessarily the same, and 𝒢2 is not connected but not zero. Then (8) achieves the desired formation if and only if 𝑣𝑝 ∕∈ ker(𝐿3 ) ∩ eig(𝐿1 𝐿2 )∖ ker(𝐿1 𝐿2 ), where 𝑣𝑝 is an eigenvector of 𝑃 (𝑠) corresponding to eigenvalue zero, ker denotes the kernel, and eig denotes the eigenvector space. Moreover, if 𝑣𝑝 ∕∈ ker(𝐿3 ) ∩ eig(𝐿1 𝐿2 )∖ ker(𝐿1 𝐿2 ), then [ T ] 0 1 11 𝑋0 lim𝑡→∞ 𝑋(𝑡) = 𝑞 0 11T ]T [ 0 ⋅ ⋅ ⋅ 0 0 𝑤12 ⋅ ⋅ ⋅ 𝑤1𝑞 + [ ] ∑𝑞 0 − 1𝑞 𝑗=2 𝑤1,𝑗 ⊗ (11) 1 Proof: Based on linear systems theory,∫ we can write 𝑡 the solution to (8) as 𝑋(𝑡) = 𝑒Φ𝑡 𝑋0 + 0 𝑒Φ(𝑡−𝜏 ) Ψ𝑑𝜏 , Φ𝑡 which includes ∫a zero input response 𝑒 𝑋0 and a nonzero 𝑡 input response 0 𝑒Φ(𝑡−𝜏 ) Ψ𝑑𝜏 . The zero input response has been considered in [20] and recall from [20] that if 𝒢1 and 𝒢2 are connected but not necessarily the same, and 𝒢3 is disconnected, then (8) achieves consensus (i.e., 𝑥𝑖 = 𝑥𝑗 ) when 𝑤𝑖,𝑗 = 0 as 𝑡 → ∞ if and only if 𝑣𝑝 ∕∈ ker(𝐿3 ) ∩ eig(𝐿1 𝐿2 )∖ ker(𝐿1 𝐿2 ). In this case, [ ] 1 11T 0 Φ𝑡 (12) = lim 𝑒 𝑡→∞ 11T 𝑞 0 To formulate the nonzero input response of the solution, note that ∫𝑡 lim𝑡→∞ 0 𝑒Φ(𝑡−𝜏 ) Ψ𝑑𝜏 ∫𝑡 = 𝑃 lim𝑡→∞ 0 𝑒𝐷(𝑡−𝜏 ) 𝑑𝜏 𝑃 −1 [ ][ ] ] 1 0 𝑤1T [ = 𝑤 1 𝑤2 0 1 𝑤2T ⎡ ⎤ 𝑒3 ( ) ] [ ⎢ .. ⎥ 1 1 + 𝑤3 ⋅ ⋅ ⋅ 𝑤2𝑞 diag 𝜆 , ⋅ ⋅ ⋅ , 𝜆 ⎣ . ⎦ (13) Φ,3
Φ,2𝑞
𝑒2𝑞
where 𝑤𝑖 is the 𝑖th column for the matrix Φ, 𝜆Φ,𝑖 is the 𝑖th eigenvalue of Φ, 𝑒𝑖 is the 𝑖th row of matrix 𝑃 , 𝑖 = 3, . . . , 2𝑞, and diag denotes a diagonal matrix. One can verify that [ [ ][ ] ] ] 1 0 𝑤1T [ 1 11T 0 𝑤 1 𝑤2 = (14) 0 1 𝑤2T 11T 𝑞 0 and [
𝑤3
⋅⋅⋅
] 𝑤2𝑞 diag
(
1972
𝜆Φ,3
,⋅⋅⋅ , [
( =
where 1 denotes a vector whose all entries are ones.
1
Φ−
)
1 𝜆Φ,2𝑞 T
1 11 𝑞 0
⎡
⎤ 𝑒3 ⎢ .. ⎥ ⎣ . ⎦
0 11T
𝑒2𝑞 ])−1 (15)
based communication/sensing cost 𝐶(𝒢) is defined as
Next, we will prove the following identity (
[
1 Φ− 0 ⎡
] )−1 [ 1 T − 𝑞 11 0 ⊗ 11T = 1 −𝑆2
𝑆1 𝐿𝑀
]
𝐶(𝒢) =
𝑖,𝑗=1
⎤ ⋅⋅⋅ −1 ⎢ .. ⎥ ⎢ −1 𝑞 − 1 . . . . ⎥ ⎢ ⎥, 𝐿1 𝑆1 = −𝐿𝑞 , where 𝐿𝑞 = ⎢ . ⎥ . . .. .. ⎣ .. −1 ⎦ −1 ⋅⋅⋅ −1 𝑞 − 1 𝐿3 𝑆2 = −𝐿𝑞 , and 𝐿2 𝑆1 = −𝐿3 𝐿𝑞 . To see this, note that ( [ ] )[ 1 T ] 𝑆1 − 𝑞 11 1 0 Φ+ ⊗ 11T 0 1 −𝑆2 𝐿𝑞 ][ [ ] T 1 −𝐿3 −𝐿2 + 𝑞 11 − 1𝑞 11𝑇 𝑆1 = T 1 𝐿1 −𝑆2 𝐿𝑞 𝑞 11 [ ] T 1 𝑞 11 + 𝐿3 𝑆2 −(𝐿2 𝑆1 + 𝐿3 𝐿𝑞 ) = 0 𝐿1 𝑆1 + 1𝑞 11T =
𝐼
𝑞−1
𝑞 ∑
−1
(16)
Next, it follows from the above identity that ∫𝑡 lim𝑡→∞ 0 𝑒Φ(𝑡−𝜏 ) Ψ𝑑𝜏 × Ψ ∫𝑡 = 𝑃 lim𝑡→∞ 0 𝑒𝐷(𝑡−𝜏 ) 𝑑𝜏 𝑃 −1 Ψ [ ][ ] ] 1 0 𝑤1T [ ×Ψ = 𝑤1 𝑤2 0 1 𝑤2𝑇 ⎡ ⎤ 𝑒3 ( ) ] [ ⎢ .. ⎥ 1 1 + 𝑤3 ⋅ ⋅ ⋅ 𝑤2𝑞 diag 𝜆Φ,3 , ⋅ ⋅ ⋅ , 𝜆Φ,2𝑞 ⎣ . ⎦ × Ψ 𝑒2𝑞 ] [ 1 T 𝑆1 − 𝑞 11 ×Ψ =0+ −𝑆2 𝐿𝑞 ] ([ ] [ 1 T 𝑆1 − 𝑞 11 −1 0 × ⊗ 𝐿3 = 0 0 −𝑆2 𝐿𝑞 [ ]T ) × 0 𝑤12 ⋅ ⋅ ⋅ 𝑤1𝑞 0 ⋅ ⋅ ⋅ 0 ] [ ]T 0 0 [ 0 𝑤12 ⋅ ⋅ ⋅ 𝑤1𝑞 0 ⋅ ⋅ ⋅ 0 = 𝐿 3 𝑆2 0 ]T [ 0 ⋅ ⋅ ⋅ 0 0 𝑤12 ⋅ ⋅ ⋅ 𝑤1𝑞 = [ ] ∑𝑞 0 − 1𝑞 𝑗=2 𝑤1,𝑗 ⊗ 1
𝑑𝑖,𝑗 𝐿𝒢𝑖,𝑗
(17)
where 𝑑𝑖,𝑗 is the geospatial distance between agents 𝑖 and 𝑗. Next, the convergence time for the multiagent system is another factor that we need to consider. To this end, let 𝐼(𝒢) denote the iteration number of numerical algorithms for the multiagent system to reach the desired formation under 𝒢 within allowed numerical error bounds. Now together with the communication/sensing cost, we introduce the following optimization problem given by minimize subject to
𝑤1 𝐶(𝒢2 ) +
𝑤2 𝐼(𝒢2 ) 𝐼(𝒢𝑐 )
𝐿2 = ∕ 0, 𝑣𝑝 ∕∈ ker(𝐿3 ) ∩ eig(𝐿1 𝐿2 )∖ ker(𝐿1 𝐿2 ) (18)
where 𝐼(𝒢𝑐 ) is the iteration number when 𝒢2 is a complete graph topology, and 𝑤1 , 𝑤2 are two positive weight constants. VI. S IMULATION R ESULTS
By superposition, we have (11). Based on Theorem 5.2, the topological condition for formation control is relaxed at certain extent, so a further question might be raised is that which topologies are the optimal ones for our multiagent system. To answer this question, we consider a similar topology optimization problem of 𝒢2 in [20] with one more distance penalty function for our observer-based formation control protocol. To measure the communication and interagent sensing cost of the multiagent system under a graph 𝒢, first the distance-
In this section, the simulation results for the proposed formation control protocol (7) are presented. Topologies for 𝒢3 and 𝒢1 are shown in Fig. 9 and 10, respectively. Here a 5-agent network is considered. In the simulation, the desired formation for the agents is the formation that each agent has the same distance “1” with each other from agent 1 to agent 5. Two disconnected cases for 𝒢2 are provided. For Case 1, Fig. 11 is the topology for 𝒢2 , and the results are shown in Fig. 13 and 14, respectively. 𝒢2 is shown as Fig. 12 in Case 2, and the states of the agents and observers versus time are shown in Fig. 15 and 16, respectively. It follows from Fig. 13 and 15 that our multiagent formation control achieves the desired formation. It is important to point out that the difference between two different topologies for 𝒢2 varies the convergence time for both Cases 1 and 2. Therefore, the following simulation will demonstrate how to solve the topology optimization problem for this multiagent formation numerically by means of the BMCO algorithm. In the second part of simulation, we seek the optimal topology for 𝒢2 to minimize the optimization problem (18) while the multiagent system achieves the desired formation. The topologies for 𝒢1 and 𝒢3 are given by Fig. 17 and 18 for five agents, respectively. The geospatial distance matrix for 𝒢2 is given by (19), 𝑤1 = 550, and 𝑤2 = 1. By running every one of the BPSO, BMCO, and NBPSO algorithms 20 times, Table II presents the min, max, median, average values for the numerical results. Moreover, Fig. 19 shows the convergence rate for all the three algorithms. By comparing the three algorithms, one can see that the BMCO algorithm surpasses the BPSO and NBPSO algorithms. The optimal solution obtained by the BMCO algorithm is shown in Fig. 20. Furthermore, based on the optimal topology for 𝒢2 ,
1973
10 8
x
1
x
2
6
x3 x4
4
States
x5 2 0 −2 −4 −6
0
5
10
15
20
25
30
35
Time
Fig. 9.
Fig. 13.
Topology for 𝒢3
Formation control for the multiagent system in Case 1
5 4 3
State
2 1 0
xc1 xc2
−1
xc3 x
c4
−2
xc5 −3
0
5
10
15
20
25
30
35
Time
Fig. 14.
Fig. 10.
Topology for 𝒢1
Observer state in Case 1
7 x1 6
x2
5
x4
x
3
x
5
States
4
the multiagent system achieves the desired formation which is presented in Fig. 21, and the observer state is shown in Fig. 22.
[𝑑𝑖,𝑗 ]𝑖,𝑗=1,2,...,5
0 ⎢775 ⎢ =⎢ ⎢422 ⎣329 758
775 0 444 1046 890
422 444 0 218 313
329 1046 218 0 104
⎤ 758 890⎥ ⎥ 313⎥ ⎥ 104⎦ 0
2 1 0 −1
0
5
10
15
20
25
Time
Fig. 15.
(19)
Formation control for the multiagent system in Case 2
5 xc1 xc2
4
x
c3
xc4
3
xc5
2
State
⎡
3
1
0
−1
VII. C ONCLUSION
−2
Case 1: topology for 𝒢2
Fig. 12.
Case 2: topology for 𝒢2
5
10
15
20
Time
In this paper, a new BMCO algorithm is proposed by introducing the communication topology for velocity and position of the particles in a BPSO-like algorithm and embedding cooperative control terms into the update formula.
Fig. 11.
0
1974
Fig. 16.
Observer state in Case 2
Fig. 17.
Topology for 𝒢1
Fig. 18.
Topology for 𝒢3
25
TABLE II C OMPARISON B ETWEEN BPSO, BMCO, AND NBPSO FOR S OLVING THE O PTIMAL F ORMATION C ONTROL P ROBLEM
4
1.5
x 10
BMCO BPSO NBPSO
1.4 1.3 1.2
f(x)
1.1 1
Min Max Median Average
0.9 0.8 0.7 0.6 0.5
0
Fig. 19.
2
4
6 8 Iteration index
10
12
Optimal topology 𝒢2∗
The efficacy and accuracy of the proposed BMCO algorithm is demonstrated by testing several benchmark optimization problems compared with the standard BPSO and a variation called NBPSO. Moreover, as an application, the topology optimization problem for an observer-based multiagent formation control protocol is formulated by considering the operation time and communication cost in the multiagent formation problem. The BPSO, BMCO, and NBPSO algorithms are used to solve the proposed topology optimization problem numerically. By evaluating the performance of all the three binary optimization algorithms, we can conclude that the proposed BMCO algorithm achieves the best optimal solution. Using the optimal solution obtained from BMCO, we also present the simulation results for the multiagent formation control protocol. R EFERENCES [1] M. M. Flood, “The traveling-salesman problem,” Operations Research, vol. 4, no. 1, pp. 61–75, 1956.
7 x1 x2
6
x3 x4
5
x5
States
4 3 2 1 0 −1
0
50
100
150
Time
Fig. 21.
Formation control for the multiagent system
5 xc1 4.5
xc2
4
xc4
xc3
States
[2] S. Kar and J. M. F. Moura, “Sensor networks with random links: Topology design for distributed consensus,” IEEE Trans. Signal Process., vol. 56, no. 7, pp. 3315–3326, 2008. [3] Z. Liu, H. Zhang, P. Smith, and Q. Hui, “Hierarchical optimization strategies for sensor network deployment,” in 2012 World Automation Congress, Puerto Vallarta, Mexico, 2012. [4] J. Kennedy and R. C. Eberhart, “A discrete binary version of the particle swarm algorithm,” in 1997 IEEE Int. Conf. Syst. Man Cybern., vol. 5, Chicago, IL, 1997, pp. 4104–4108. [5] J. Sadri and C. Y. Suen, “A genetic binary particle swarm optimization model,” in 2006 IEEE Congr. Evolut. Comput., Vancouver, Canada, 2006, pp. 656–663. [6] G. Pampara, N. Franken, and A. P. Engelbrecht, “Combining particle swarm optimisation with angle modulation to solve binary problems,” in 2005 IEEE Congr. Evolut. Comput., Edinburgh, UK, 2005, pp. 89–96. [7] M. A. Khanesar, M. Teshnehlab, and M. A. Shoorehdeli, “A novel binary particle swarm optimization,” in 15th IEEE Med. Conf. Control Automation, Athens, Greece, 2007, pp. 1–6. [8] Q. Hui, W. M. Haddad, and S. P. Bhat, “Finite-time semistability and consensus for nonlinear dynamical networks,” IEEE Trans. Autom. Control, vol. 53, pp. 1887–1900, 2008. [9] ——, “Semistability, finite-time stability, differential inclusions, and discontinuous dynamical systems having a continuum of equilibria,” IEEE Trans. Autom. Control, vol. 54, no. 10, pp. 2465–2470, 2009. [10] Q. Hui, “Hybrid consensus protocols: An impulsive dynamical system approach,” Int. J. Control, vol. 83, pp. 1107–1116, 2010. [11] P. N. Suganthan, “Particle swarm optimiser with neighbourhood operator,” in 1999 IEEE Congr. Evolut. Comput., 1999, pp. 1958–1962. [12] D. Parrott and X. Li, “Locating and tracking multiple dynamic optima by a particle swarm model using speciation,” IEEE Trans. Evolut. Comput., vol. 10, no. 4, pp. 440–458, 2006. [13] F. Xiao, L. Wang, J. Chen, and Y. Gao, “Finite-time formation control for multi-agent systems,” Automatica, vol. 45, no. 11, pp. 2605–2611, 2009. [14] Y. Hong, J. Hu, and L. Gao, “Tracking control for multi-agent consensus with an active leader and variable topology,” Automatica, vol. 42, no. 7, pp. 1177–1182, 2006. [15] N. Sorensen and W. Ren, “A unified formation control scheme with a single or multiple leaders,” in Proc. Amer. Control Conf., New York, NY, 2007, pp. 5412–5418. [16] W. Ren and E. Atkins, “Distributed multi-vehicle coordinated control via local information exchange,” Int. J. Robust Nonlinear Control, vol. 17, no. 10-11, pp. 1002–1033, 2007. [17] W. Ren, “Consensus based formation control strategies for multi-vehicle systems,” in Proc. Amer. Control Conf., Minneapolis, MN, 2006, pp. 4237–4242. [18] H. Zhang, S. Mullen, and Q. Hui, “A note to robustness analysis of the hybrid consensus protocols,” in Proc. Amer. Control Conf., San Francisco, CA, 2011, pp. 1088–1093. [19] Q. Hui and W. M. Haddad, “Distributed nonlinear control algorithms for network consensus,” Automatica, vol. 44, pp. 2375–2381, 2008. [20] H. Zhang and Q. Hui, “Topological heterogeneity and optimality analysis for multiagent formation,” in Proc. IEEE Conf. Decision Control, Maui, HI, 2012, pp. 5954–5959.
xc5
3.5 3 2.5 2 1.5 1 0.5
0
50
100
150
Time
Fig. 22.
NBPSO 4.07E3 8.02E3 5.69E3 5.79E3
14
Comparison between the three algorithms
Fig. 20.
BPSO BMCO 4.33E3 4.21E3 8.17E3 7.64E3 6.05E3 5.10E3 6.24E3 5.62E3
Observer state
1975
