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Nonuniform coverage control for heterogeneous mobile sensor networks on the line
Automatica 81 (2017) 464–470
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Automatica journal homepage: www.elsevier.com/locate/automatica
Brief paper
Nonuniform coverage control for heterogeneous mobile sensor networks on the line✩ Liya Dou a,b , Cheng Song c , Xiaofan Wang a , Lu Liu b,1 , Gang Feng b a
Department of Automation, Shanghai Jiao Tong University, Shanghai 200240, China
b
Department of Mechanical and Biomedical Engineering, City University of Hong Kong, Kowloon, Hong Kong Special Administrative Region
c
School of Automation, Nanjing University of Science and Technology, Nanjing 210094, China
article
info
Article history: Received 16 May 2016 Received in revised form 21 February 2017 Accepted 14 March 2017
Keywords: Nonuniform coverage Mobile sensor network Distributed control Actuation constraint
abstract The coverage control problem for networked mobile sensors on a line with different actuation limits is addressed in this paper. The roughness of each point on the line is assumed to be different which makes the concerned problem more challenging. The objective of coverage control considered in this paper is to minimize the largest time required for the sensor network to reach any point on the line via optimizing the sensors’ locations on this line. Distributed coverage control laws with input constraints are developed to drive the sensors to the optimal configuration while preserving their spatial ordering on the line. Simulation examples demonstrate the effectiveness of the proposed control laws. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction Networks of sensors and autonomous vehicles have wide applications in many fields, including environment monitoring, surveillance, industrial diagnostics and so on. When the mission domain is remote and/or hostile, such networks must be capable of operating in a fully autonomous and distributed way to adapt to changing environments. The potential of such applications has prompted much interest in coverage control of mobile sensor networks, where the goal is to drive networked mobile sensors to an optimal sensing configuration in a mission field. If the information density or terrain roughness is uniform over the mission field, the problem is referred to as the uniform coverage control problem, whereas for a nonuniform field, the problem is referred to as the nonuniform coverage control problem with a nonuniform metric.
✩ This work was supported by grants from the Research Grants Council of Hong Kong (Nos. CityU-11213415, CityU-11261516), the National Natural Science Foundation of China (Nos. 61374176, 61403203), and the Natural Science Foundation of Jiangsu Province (No. BK20140798). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Michael M. Zavlanos under the direction of Editor Christos G. Cassandras. E-mail addresses: [email protected] (L. Dou), [email protected] (C. Song), [email protected] (X. Wang), [email protected] (L. Liu), [email protected] (G. Feng). 1 Fax: +852 3442 0172.
http://dx.doi.org/10.1016/j.automatica.2017.04.029 0005-1098/© 2017 Elsevier Ltd. All rights reserved.
In practice, mobile sensors may have some physical constraints like limited communication or sensing ranges (Laventall & Cortés, 2009), velocity or actuation constraints (Braun et al., 2013; Brezak & Petrovic, 2011). Accordingly, a network of heterogeneous mobile sensors in terms of their different actuation limits is considered in this work. This paper investigates the nonuniform coverage control of a network of heterogeneous mobile sensors located on a line with varying roughness. 1.1. Related work There is a large volume of works on coverage control problems of sensor networks in the open literature, see for examples Bartolini, Calamoneri, La Porta, and Silvestri (2011), Breitenmoser, Schwager, Metzger, Siegwart, and Rus (2010), Caicedo-Nuez and Žefran (2008), Cortés and Bullo (2005), Cortés, Martınez, and Bullo (2005); Cortés, Martınez, Karatas, and Bullo (2004), Kantaros, Thanou, and Tzes (2015), Laventall and Cortés (2009), Li and Cassandras (2005), MartíNez and Bullo (2006), Pimenta, Kumar, Mesquita, and Pereira (2008), Poduri and Sukhatme (2004), Song, Liu, Feng, Wang, and Gao (2013), Zhai and Hong (2013) and references therein. Based on Voronoi partition and gradient descent algorithms, coverage control laws are developed for a group of mobile sensors in uniform fields in Cortés and Bullo (2005). Uniform coverage control with a constraint on the minimum number of each agent’s neighbors is investigated in Poduri and Sukhatme (2004).
L. Dou et al. / Automatica 81 (2017) 464–470
Nonuniform coverage control of sensor networks is addressed in part by using density-dependent gradient descent laws in Cortés et al. (2004), Li and Cassandras (2005) and Pimenta et al. (2008). Nonuniform coverage control of mobile agents with limited sensing or communication range is treated in Cortés et al. (2005), Kantaros et al. (2015) and Laventall and Cortés (2009). The authors in Lekien and Leonard (2009) investigate the nonuniform coverage problem of a planar region with a non-Euclidean distance metric that stretches and shrinks regions of high and low density, respectively. Nonuniform coverage control laws are proposed using cartogram transformation which requires some global information about the region. In addition, nonuniform coverage of a planar region by a group of mobile sensors with stochastically intermittent communications among them is studied in Miah, Nguyen, Bourque, and Spinello (2015). 1.2. One-dimensional coverage problem This paper focuses on the nonuniform coverage problem when the sensors are arranged on a line, which is motivated by two considerations. First, the one-dimensional coverage problem has several potential applications such as environmental boundary monitoring and target tracking (Jin & Bertozzi, 2007; Susca, Bullo, & Martínez, 2008), and it has attracted increasing attention in recent years, such as Choi and Horowitz (2010), Davison, Leonard, Olshevsky, and Schwemmer (2015), Frasca, Garin, Gerencsér, and Hendrickx (2015). When a group of mobile sensors performs border patrol, they should be optimally positioned on a curve. This problem can be transformed to the line coverage problem investigated in this paper by parametrizing the curve with its arc-length. Second, the one-dimensional coverage problem can provide a simplified setting for addressing unsettled issues in coverage control. One typical issue is to design control algorithms for mobile agents to minimize the response time or energy cost from the agents to any point in the mission field. The paper Leonard and Olshevsky (2013) develops a distributed coverage control law for a nonuniform field under which a group of mobile agents are optimally positioned on a line, such that the distance from the mobile agents to any point on the line is minimized. Taking into consideration the sensors’ velocity constraints, the authors in Song, Liu, Feng, and Xu (2016) study the uniform coverage problem of a circle by networked mobile sensors to minimize the largest time taken from the network to any point on the circle. 1.3. Our contributions In this paper, we consider the nonuniform coverage control of a network of heterogeneous mobile sensors with different actuation limits located on a line. The roughness of each point on the line is assumed to be different. Our objective is to minimize a coverage cost function, defined as the largest time required for the sensor network to reach any point on the line. Distributed control laws are designed, under which the sensors are driven to the optimal positions with order preservation. Compared with Leonard and Olshevsky (2013), where nonuniform coverage control laws are designed for homogeneous sensors without actuation constraints, this paper takes into account different actuation limits of mobile sensors and varying roughness of the terrain, and designs distributed control laws with input saturation. The problem investigated in this paper is more general. Compared with Song et al. (2016), where coverage control problem of networked mobile sensors with different velocity constraints positioned on a circle is studied, this paper investigates the coverage problem of mobile sensors with actuation constraints located on a line, and also takes varying roughness of the terrain into consideration.
465
The remainder of this paper is structured as follows. Section 2 formalizes the nonuniform coverage problem with actuation constraints. Distributed coverage control laws are proposed in Section 3. Section 4 presents the convergence analysis of the coverage control laws. In Section 5, the effectiveness of the results is illustrated by simulations. Finally, Section 6 concludes the paper. 2. Problem formulation Consider a network of n mobile sensors initially located at arbitrary positions q1 (0), q2 (0), . . . , qn (0) which are assumed, without loss of generality, to lie in the interval [0, 1]. Denote R+ as the set of positive real numbers. Let ρ : [0, 1] → R+ , be a positive, piecewise-continuous function, which measures the roughness at each point on the line. We assume that ρ is bounded, i.e., there exist positive constants ρmax and ρmin such that for all z ∈ [0, 1], we have ρmin ≤ ρ(z ) ≤ ρmax . In this work, the roughness function ρ is assumed to be known a priori by the sensors. Following Leonard and Olshevsky (2013), the distance between a and b ∈ [0, 1] is defined as dρ (a, b) =
max(a,b) min(a,b)
ρ(z )dz ,
b ρ(z )dz. It is easy to see that and we denote d¯ ρ (a, b) = a dρ (a, b) = |d¯ ρ (a, b)|. We label the sensors from 1 to n in accordance with their initial order along the line from left to right and assume that no two sensors occupy the same position, that is,
0 < q1 (0) < q2 (0) < · · · < qn (0) < 1.
(1)
We introduce the function F (x) =
x
ρ(z )dz . 0
Note that F (1) = dρ (0, 1). Furthermore, for any two points a < b in [0, 1], dρ (a, b) = F (b) − F (a). In this paper, we consider the nonuniform coverage problem for heterogeneous mobile sensors with actuation constraints. There exists a maximum actuation αi for each sensor. Networked mobile sensors are assumed to evolve in discrete time periods. The maximum distance that each sensor can move in one sampling period is limited, which depends on both the sensor’s actuation limit and the varying roughness of the terrain. The actuation needed q for each agent to move from point p to q on the line is given by | p f (ρ(z ))dz |, where f (x) : R+ → R+ is a strictly increasing continuous function with respect to x. This definition captures the fact that the agents need larger actuation for movement in areas where the roughness function ρ is higher, and smaller actuation in areas where ρ is lower. For simplicity, we assume f (ρ(z )) = ρ(z ) in this work. As sensor i has the maximum actuation αi , the actuation for sensor i to move on the line in one period is no larger
q (k+1)
than αi , that is, −τ αi ≤ q i(k) ρ(z )dz ≤ τ αi for i = 1, . . . , n, i and k ≥ 0, where qi (k) is the position of sensor i on the line at the discrete-time index k and τ > 0 is the sampling period. According to the above considerations, the evolution of the mobile sensors’ positions depends on not only the control inputs but also the varying roughness of the terrain. For our analysis, each mobile sensor is assumed to evolve according to
qi (k+1) qi (k)
ρ(z )dz = τ ui (k),
(2)
where ui (k) is the control input of sensor i at the discrete-time index k. Since τ > 0 is the sampling period, ui (k) also can be regarded
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L. Dou et al. / Automatica 81 (2017) 464–470
as the actuation of sensor i moving on the line with varying roughness at the discrete-time index k. Due to the fact that the sensors have different actuation limits, it is reasonable to consider control inputs with different constraints, that is, −αi ≤ ui (k) ≤ αi , k ≥ 0, i = 1, . . . , n. In order to make the networked mobile sensors’ dynamics easy to analyze, we define yi (k) = F (qi (k)), i = 1, . . . , n. From (2), we can conclude that the variables yi (k) evolve as follows, yi (k + 1) = yi (k) + τ ui (k),
i = 1 , . . . , n.
(3)
We assume that there exists an upper bound αM on the sensors’ actuation limits and it is known by the networked mobile sensors beforehand. To simplify notation, let q0 (k) = 0, qn+1 (k) = 1, y0 (k) = 0 and yn+1 (k) = F (1) for all k, and α0 = αn+1 = 0 throughout the paper. We define the coverage cost function as T (q1 , . . . , qn , ρ) = max min
q∈[0,1] i=1,...,n
dρ (qi , q)
αi
,
where dρ (qi , q) is the weighted distance between sensor i and point q on a line with varying roughness. The time required for mobile sensor i to move to point q on the line depends on the weighted distance dρ (qi , q) and the actuation limit of sensor i. From (2), we know that αi denotes the maximum weighted distance that sensor i can move in one sampling period. Therefore, dρ (qi , q)/αi indicates the shortest time required for sensor i to arrive at point q. Define the time required for a mobile sensor network to reach a point on the line as the minimum of the shortest time required for the sensors to arrive at the point. As a result, T represents the largest time required for the network to reach any point on the line. Our objective is to drive networked mobile sensors to the optimal coverage configuration, that is, arg minq1 ,...,qn T (q1 , . . . , qn , ρ), by designing distributed control laws. 3. Distributed coverage control laws
i = 1 , . . . , n,
2 Lemma 1. Suppose condition (1) is satisfied. If 0 < τ < 1/(4αM ), then with the coverage control laws (7), 0 < q1 (k) < q2 (k) < · · · < qn (k) < 1 holds for all k ≥ 0.
Proof. First we prove that the ordering of the sensors is preserved by using the following control inputs without saturation u˜ i (k) = αi u¯ i (k),
u¯ 1 (k) = α1 d¯ ρ (q1 (k), q2 (k)) − (α1 + α2 )d¯ ρ (0, q1 (k)), u¯ i (k) = (αi−1 + αi )d¯ ρ (qi (k), qi+1 (k)) − (αi + αi+1 ) ×d¯ ρ (qi−1 (k), qi (k)), i = 2, . . . , n − 1, ¯un (k) = (αn−1 + αn )d¯ ρ (qn (k), 1) − αn d¯ ρ (qn−1 (k), qn (k)).
(5)
yi (k + 1) = yi (k) + τ αi u¯ i (k),
i = 1, . . . , n.
Define h0 (k) = y1 (k), hi (k) = yi+1 (k) − yi (k), hn (k) = F (1) − yn (k).
i = 1, . . . , n − 1,
The control laws u¯ i , i = 1, . . . , n can be rewritten in the following form u¯ 1 (k) = α1 h1 (k) − (α1 + α2 )h0 (k), u¯ i (k) = (αi−1 + αi )hi (k) − (αi + αi+1 )hi−1 (k),
i = 2, . . . , n − 1, (11)
u¯ n (k) = (αn−1 + αn )hn (k) − αn hn−1 (k).
Recall that α0 = αn+1 = 0, y0 (k) = 0 and yn+1 (k) = F (1). Therefore, under the control laws (10), if hi (k) > 0, and hi−1 (k) > 0, i = 1, . . . , n, one has
= τ αi [(αi−1 + αi )hi (k) − (αi + αi+1 )hi−1 (k)] ≤ 2τ αM2 hi (k), i = 1, . . . , n. Similarly, i = 0, . . . , n − 1.
The above two inequalities imply that yi+1 (k + 1) − yi+1 (k) − (yi (k + 1) − yi (k))
(6)
We can rewrite compactly the control laws (4) as u(k) = α sat(¯u(k)),
(10)
2 yi+1 (k + 1) − yi+1 (k) ≥ −2τ αM hi (k),
It is noted that for i = 2, . . . , n − 1, u¯ i makes use of the weighted distance over the roughness ρ from sensor i to sensors i − 1 and i + 1, that is, d¯ ρ (qi−1 (k), qi (k)) and d¯ ρ (qi (k), qi+1 (k)), respectively. It means that sensor i coordinates its motion with that of sensors i − 1 and i + 1 for i = 2, . . . , n − 1. Particularly, sensor 1 updates its position based on the distance from it to the border 0 and sensor 2, and sensor n updates its position in a similar way. By the definition of yi (k), Eq. (5) can be equivalently written as u¯ 1 (k) = α1 (y2 (k) − y1 (k)) − (α1 + α2 )(y1 (k) − 0), u¯ i (k) = (αi−1 + αi )(yi+1 (k) − yi (k)) − (αi + αi+1 ) ×(yi (k) − yi−1 (k)), i = 2, . . . , n − 1, u¯ n (k) = (αn−1 + αn )(F (1) − yn (k)) − αn (yn (k) − yn−1 (k)).
i = 1 , . . . , n,
where u¯ i (k) is given by Eq. (6). In this case, the mobile sensors evolve as follows,
(4)
where sat(·) : R → [−1, 1] is a standard saturation function defined as sat(x) = sign(x) min{1, |x|}, and
(7)
where u(k) = [u1 (k), u2 (k), . . . , un (k)] , α is a diagonal matrix with the ith diagonal element being αi , and u¯ (k) = [¯u1 (k), u¯ 2 (k), . . . , u¯ n (k)]T is given by T
u¯ (k) = −Ly(k) + B
Collisions can be avoided during the coverage task if the ordering of the mobile sensors is preserved. The following lemma shows that the execution of the coverage control laws (7) preserves the sensors’ ordering.
yi (k + 1) − yi (k) = τ αi u¯ i (k)
We propose the coverage control laws as follows ui (k) = αi sat(¯ui (k)),
with y(k) = [y1 (k), y2 (k), . . . , yn (k)]T , B = [0, . . . , 0, (αn−1 + αn )F (1)]T and L is given by Eq. (9) in Box I.
(8)
≥ −4τ αM2 hi (k),
i = 0, 1, . . . , n,
that is 2 hi (k + 1) ≥ (1 − 4τ αM )hi (k),
i = 0, 1, . . . , n,
2 which is larger than zero provided that 0 < τ < 1/(4αM ). Consequently, yi+1 (k + 1) > yi (k + 1) if yi+1 (k) > yi (k) for i = 0, 1, . . . , n. Since F (x) is a strictly monotonically increasing function, it is equivalent to say that qi+1 (k + 1) > qi (k + 1) if qi+1 (k) > qi (k) for i = 0, 1, . . . , n. Thus, with the control laws (10) the ordering of the sensors is preserved. Then we show that the coverage control laws (7) with saturation also preserve the ordering of the sensors. With the control laws (7), we can obtain
h0 (k + 1) = h0 (k) + τ α1 sat(¯u1 (k)), hi (k + 1) = hi (k) + τ (αi+1 sat(¯ui+1 (k)) − αi
× sat(¯ui (k))), i = 1, . . . , n − 1, hn (k + 1) = hn (k) − τ αn sat(¯un (k)).
(12)
L. Dou et al. / Automatica 81 (2017) 464–470
2α + α 1 2 .. . L = ··· .. . 0
−α1 .. .
··· .. .
0
467
0
− (αi + αi+1 ) .. .
.. . αi−1 + 2αi + αi+1 .. .
− (αi−1 + αi ) .. .
0
···
−αn
.. . ··· .. .
.
(9)
αn−1 + 2αn
Box I.
We will prove that given the sampling period 0 < τ < 2 1/(4αM ), hi (k+1) > 0 holds if hi (k) > 0 for i = 1, . . . , n−1. When u¯ i+1 (k) ≥ 1 and u¯ i (k) < −1, hi (k + 1) = hi (k) + τ (αi+1 + αi ) > 0 if hi (k) > 0. When u¯ i+1 (k) ≥ 1 and −1 ≤ u¯ i (k) ≤ 1, hi (k + 1) = hi (k) + τ (αi+1 − αi u¯ i (k))
= τ αi+1 + (1 − τ αi (αi−1 + αi ))hi (k) + τ αi (αi + αi+1 )hi−1 (k) > 0 2 if hi−1 (k) > 0, hi (k) > 0 and 0 < τ < 1/(4αM ). When u¯ i+1 (k) ≥ 1 and u¯ i (k) > 1, hi (k + 1) = hi (k) + τ (αi+1 − αi ). Since u¯ i (k) = (αi−1 + αi )hi (k) − (αi + αi+1 )hi−1 (k) ≥ 1 and hi−1 (k) > 0, we have that hi (k) > 1/(αi−1 + αi ). Therefore, hi (k + 1) > 1/(αi−1 + αi ) + τ (αi+1 − αi ) > 0 holds when 2 0 < τ < 1/(4αM ). When u¯ i+1 (k) ≤ −1 and u¯ i (k) ≤ −1, hi (k + 1) = hi (k) + τ (αi − αi+1 ). Similarly, it can be shown that in 2 this case hi (k + 1) > 0 if hi+1 (k) > 0 and 0 < τ < 1/(4αM ). When −1 ≤ u¯ i+1 (k) ≤ 1 and u¯ i (k) ≤ −1,
hi (k + 1) = hi (k) + τ (αi+1 u¯ i+1 (k) + αi )
= τ αi + (1 − τ αi+1 (αi+1 + αi+2 ))hi (k) + τ αi+1 (αi + αi+1 )hi+1 (k) > 0 2 if 0 < τ < 1/(4αM ), hi (k) > 0 and hi+1 (k) > 0. When u¯ i+1 (k) < 1 and u¯ i (k) ≥ −1, we can obtain that hi (k + 1) ≥ hi (k) + τ (αi+1 u¯ i+1 (k) − αi u¯ i (k)) > 0 holds since it has been proved that hi (k + 1) = hi (k) + τ (αi+1 u¯ i+1 (k) − αi u¯ i (k)) > 0 if hi (k) > 0 for i = 1, . . . , n − 1 under the coverage control laws (10). For h0 (k), we note that h0 (k + 1) > h0 (k) + τ α1 u¯ 1 (k) when u¯ 1 (k) < 1. As we have shown that h0 (k + 1) = h0 (k) + τ α1 u¯ 1 (k) > 0 if h0 (k) > 0, h0 (k + 1) > 0 holds if h0 (k) > 0 when u¯ 1 (k) < 1. When u¯ 1 (k) ≥ 1, it is easy to see that h0 (k + 1) = h0 (k)+τ α1 > 0 if h0 (k) > 0. Similarly, we can prove that hn (k + 1) > 0 if hn (k) > 0. In conclusion, hi (k + 1) > 0 if hi (k) > 0 for i = 0, 1, . . . , n with the control laws (7). If condition (1) is satisfied and 0 < τ < 2 1/(4αM ), with the proposed coverage control laws (7), 0 < q1 (k) < · · · < qi (k) < qi+1 (k) < · · · < qn (k) < 1 holds for all k ≥ 0.
Remark 1. Under the given conditions in Lemma 1, 0 < q1 (k) < q2 (k) < · · · < qn (k) < 1 holds for all k ≥ 0. In this case, d¯ ρ (qi (k), qi+1 (k)) = dρ (qi (k), qi+1 (k)) for i = 0, . . . , n, and k ≥ 0. From (5), one has −2αM dρ (qi−1 (k), qi (k)) < u¯ i (k) < 2αM dρ (qi (k), qi+1 (k)). Since ui (k) = αi sat(¯ui (k)), ui (k) ≤ αi u¯ i (k) when u¯ i (k) ≥ 0, and ui (k) ≥ αi u¯ i (k) when u¯ i (k) < 0. Thus, −2αM2 dρ (qi−1 (k), qi (k)) < ui (k) < 2αM2 dρ (qi (k), qi+1 (k)). Since 2 0 < τ < 1/(4αM ) and ρ(z ) > 0, according to Eq. (2), we have
qi (k)
qi (k+1)
qi+1 (k)
ρ(z )dz < qi (k) ρ(z )dz < ρ(z )dz, and qi (k) consequently qi−1 (k) < qi (k + 1) < qi+1 (k) for i = 1, . . . , n. that −
1 2
qi−1 (k)
1 2
Note also that each mobile sensor’s neighbors are fixed since the execution of the control laws (7) preserves the sensors’ ordering. When implementing the control laws (7), sensor i, i = 2, . . . , n−1 communicates with its left neighbor and right neighbor to acquire the information of their positions and actuation limits. Then it will compute the weighted distance dρ (qi−1 (k), qi (k)) to its left neighbor and dρ (qi (k), qi+1 (k)) to its right neighbor. Based on all the information, sensor i can compute the control law ui (k), and
moves to the position in the interval (qi−1 (k), qi+1 (k)) satisfying Eq. (2). Sensors 1 and n are border sensors with a single neighbor. Sensor 1 computes its control law based on the weighted distance to its right neighbor and the left boundary of the line. Sensor n can implement the control law un (k) in a similar way. Therefore, the proposed coverage control laws (7) can be executed in a distributed way without knowledge of the sensors’ labels. 4. Convergence analysis In this section, we show that with the proposed control laws (7) a heterogeneous mobile sensor network will converge to the optimal set of positions which corresponds to the minimum coverage cost. We first give necessary and sufficient conditions for achieving the minimum of the function T (q1 , . . . , qn , ρ). Lemma 2. The coverage cost function T (q1 , . . . , qn , ρ) is minimized if and only if dρ (0, q1 )
α1
=
dρ (q1 , q2 )
= ··· =
α1 + α2
dρ (qn−1 , qn )
αn−1 + αn
=
dρ (qn , 1)
αn
. (13)
Moreover, the minimum of the function T (q1 , . . . , qn , ρ) is F (1)/ (2 nl=1 αl ). Proof. Define ρ
Vi =
q ∈ [0, 1]|
dρ (qi , q)
≤
αi
dρ (qj , q)
αj
, ∀j ̸= i .
(14)
x
ρ(z )dz is monotonically increasAs the function F (x) = 0 ρ ing, each set Vi , i = 1, . . . , n is a closed set.We can also ρ = [0, 1]. Therefore, ni=1 V ρ ρ(z )dz ≥ have that i=1,...,n Vi i
F (1). From the definition of the function T (q1 , . . . , qn , ρ) and ρ the sets Vi , i = 1, . . . , n, we have T (q1 , . . . , qn , ρ) = dρ (qi ,q)
d ( q ,q )
= maxi=1,...,n maxq∈V ρ ρ αii . Note i n ρ ρ that F (1) ≤ i=1 V ρ(z )dz ≤ 2 i=1 maxq∈Vi dρ (qi , q) ≤ i n n 2T i=1 αi . Therefore, T (q1 , . . . , qn , ρ) ≥ F (1)/(2 l=1 αl ) almaxq∈[0,1] mini=1,...,n
n
αi
ways holds. (Sufficiency) Recall that q0 (k) = 0, qn+1 (k) = 1, and α0 = αn+1 = 0. From the definition of the function T (q1 , . . . , qn , ρ), we have T (q1 , . . . , qn , ρ) = max min
dρ (qi , q)
αi
q∈[0,1] i=1,...,n
= max
max
min
dρ (qi , q)
αi
i=0,...,n q∈[qi ,qi+1 ] i=1,...,n
≤ max
max
dρ (q, q1 )
q∈[q0 ,q1 ]
α1
max
max min i=1,...,n−1 q∈[qi ,qi+1 ] max
q∈[qn ,qn+1 ]
dρ (qn , q)
αn
, dρ (qi , q) dρ (q, qi+1 )
αi
.
,
αi+1
,
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L. Dou et al. / Automatica 81 (2017) 464–470
d q,q ρ , ρ (αi+1i+1 ) , i = 1, . . . , n − 1, d0 = αi dρ (q,q1 ) dρ (qn ,q) ρ , and dn = . For an arbitrary point located in αi αn ρ [qi , qi+1 ], we have (αi + αi+1 )di ≤ dρ (qi , q) + dρ (q, qi+1 ) = ρ dρ (qi , qi+1 ), i = 1, . . . , n − 1, α1 d0 = dρ (q, q1 ) ≤ dρ (q0 , q1 ), ρ ρ and αn dn = dρ (qn , q) ≤ dρ (qn , qn+1 ). It implies that di ≤
ρ
Denote di = min
dρ (qi ,q)
dρ (qi , qi+1 )/(αi + αi+1 ), i = 0, . . . , n holds for all points located in [qi , qi+1 ]. Therefore, T (q1 , . . . , qn , ρ) ≤ maxi=0,...,n ρ maxq∈[qi ,qi+1 ] di ≤ maxi=0,...,n dρ (qi , qi+1 )/(αi + αi+1 ). When dρ (qi , qi+1 )/(αi + αi+1 ) is identical for i = 0, . . . , n, we can obn tain dρ (qi , qi+1 )/(αi + αi+1 ) = F (1)/(2 l=1 αl ), i = 0, . . . , n, n = F (1). As we have shown that by the fact i=0 dρ (qi , qi+1 ) n T (q1 , . . . , qn , ρ) ≥ F (1)/(2 l=1 αl ) holds, T (q1 , . . . , qn , ρ) = n F (1)/(2 l=1 αl ) when dρ (qi , qi+1 )/(αi + αi+1 ) is identical. (Necessity) If the function T (q1 , . . . , qn , ρ) is minimized, dρ (qi ,q)
T (q1 , . . . , qn , ρ) = maxi=1,...,n maxq∈V ρ i
αi
= F (1)/(2
n
l =1
d (q ,q) αl ). Then, maxq∈V ρ ρ αii ≤ F (1)/(2 nl=1 αl ). In consequence, i n ρ ρ(z )dz ≤ 2 maxq∈V ρ dρ (qi , q) ≤ F (1)αi / l=1 αl holds for V i
i
ρ
i = 1, . . . , n. If there exists a set Vj such that maxq∈V ρ j n n ρ ρ(z )dz < F (1)/(2 l=1 αl ), we have that i =1 V contradicts with the fact
n i =1
maxq∈V ρ dρ (qi , q) = F (1)αi /(2 i
dρ (qj ,q)
αi
F (1). It
i
ρ
Vni
<
ρ(z )dz ≥ F (1). Therefore,
αl ) for i = 1, . . . , n. Define qi qr Wi = [ , ], i = 1, . . . , n with l ρ(z )dz = q i ρ(z )dz = q i qli
l=1
qri
i
ρ
maxq∈V ρ dρ (qi , q), and we have that Vi ⊆ Wi . Combining with the i
ρ = [0, 1], it can be obtained that [0, 1] ⊆ i=1,...,n ρ(z )dz ≥ F (1). Note that maxq∈V ρ dρ (qi , q) = i ( )α /( α )holds for i = 1, . . . , n. From the definition of n Wi , we have that ρ( z ) dz = F ( 1 ) . Thus, i=1 Wi i=1,...,n Wi = [0, 1] and the interiors of the n closed sets W are disjoint. Recall i ρ ρ that the sets Vi , i = 1, . . . , n are closed sets, i=1,...,n Vi = [0, 1] ρ ρ and Vi ⊆ Wi . Therefore, Vi = Wi for all i = 1, . . . , n. As a n result, dρ (qi , qi+1 )/(αi + αi+1 ) = F (1)/(2 l=1 αl ) holds for all i = 0, . . . , n.
fact
i=1,...,n Vi n Wi and i = 1 Wi n F 1 i 2 l =1 l
Before we show that the optimality condition proposed in Lemma 2 can be satisfied as time goes to infinity, the following lemma will be introduced. Lemma 3 (Ren & Beard, 2005). Let A be a stochastic matrix with positive diagonal elements. If the graph corresponding to A contains a spanning tree, then A is indecomposable and aperiodic (SIA), that is, limk→∞ Ak = 1v T , where v satisfies AT v = v and 1T v = 1. Furthermore, each element of v is nonnegative. 2 Lemma 4. Suppose condition (1) is satisfied. If 0 < τ < 1/(4αM ),
with the control laws (7),
dρ (qi (k),qi+1 (k))
αi +αi+1
, i = 0, 1, . . . , n will con-
verge to the same limit. Proof. From the definition of u¯ (k), we have u¯ (k + 1) − u¯ (k) = −L(y(k + 1) − y(k)) = −τ α Lsat(¯u(k)).
(15)
First it can be proved that for any i = 1, . . . , n, u¯ i (k + 1) ≥ −1 if u¯ i (k) ≥ −1. Note that when u¯ i (k) ≥ −1, sat(¯ui (k)) ≤ u¯ i (k). Then we have
u¯ i (k + 1) = u¯ i (k) − τ αi Lii sat(¯ui (k)) +
Lij sat(¯uj (k))
where Lij denotes the (i, j) term of the matrix L. As Lij ≤ 0, ∀j ̸= i 2 and 1 − τ αi Lii > 0 for i = 1, . . . , n, when 0 < τ < 1/(4αM ), we n ¯ have that ui (k + 1) ≥ −1 + τ αi Lii + τ αi j̸=i Lij . Since j=1 Lij ≥ 0 for i = 1, . . . , n, u¯ i (k + 1) ≥ −1 holds. Similarly, we can prove that u¯ i (k + 1) ≤ 1 if u¯ i (k) ≤ 1. Then we will prove that u¯ (k) will in finite time enter the set Ω = {¯u ∈ Rn : |¯ui (k)| ≤ 1, i = 1, . . . , n}. For any i = 1, . . . , n, |¯ui (k + 1)| ≤ 1 if |¯ui (k)| ≤ 1. Therefore, if |¯ui (0)| ≤ 1 for all i = 1, . . . , n, |¯ui (k)| ≤ 1, ∀k > 0 holds for all i = 1, . . . , n. Ω is an invariant set of system (15). We will prove by contradiction that in finite time u¯ i (k), i = 1, . . . , n will be less than 1. Assume that there are p elements of u¯ (k) larger than 1 for all k ≥ 0, that is, u¯ l (k) > 1, l ∈ {1, . . . , n}, l = l1 , . . . , lp for k ≥ 0. According to the mobile sensors’ dynamics (3), we have that yl (k + 1) = yl (k) + τ αl ,
l = l1 , . . . , lp .
Then, yl (k) = yl (0) + τ αl k, l = l1 , . . . , lp for k ≥ 0. Since τ and αl are both positive constants, we can conclude that yl (k) → ∞ as k → ∞. It contradicts with the fact yl (k) < F (1), ∀l ∈ {1, . . . , n} from Lemma 1. Therefore, u¯ l (k), l = l1 , . . . , lp will be less than 1 at some finite discrete-time index kl . As we have shown that for any i = 1, . . . , n, u¯ i (k + 1) ≥ −1 if u¯ i (k) ≥ −1, u¯ l (k), l = l1 , . . . , lp will enter the interval [−1, 1] at the discrete-time index kl . Upon setting ks = max{kl1 , . . . , klp }, we have that all u¯ l (k), ∀l = l1 , . . . , lp will enter the interval [−1, 1] at the finite discrete-time index ks . Similarly, we can show that in finite time all the elements of u¯ which are less than −1 will enter the interval [−1, 1]. Thus, u¯ (k) will in finite time enter the invariant set Ω . 2 Finally, we will prove that if 0 < τ < 1/(4αM ) and u¯ (k) belongs h (k)
i = c, to the set Ω at a finite discrete-time index km , limk→∞ α +α i i+1 i = 0, 1, . . . , n where c is a positive number. Note that u¯ (k) ∈ Ω holds for all k ≥ km . Then we have u(k) = α u¯ (k), k ≥ km . Denote hi (k) h¯ i (k) = α +α for i = 0, 1, . . . , n. From Eqs. (11) and (12), it can i
i+1
be derived that h¯ 0 (k + 1) − h¯ 0 (k) = τ u¯ 1 (k)
= τ α1 (α1 + α2 )(h¯ 1 (k) − h¯ 0 (k)), τ (αi+1 u¯ i+1 (k) − αi u¯ i (k)) h¯ i (k + 1) − h¯ i (k) = αi + αi+1 = τ [αi+1 (αi+1 + αi+2 )h¯ i+1 (k) − (αi+1 (αi+1 + αi+2 ) + αi (αi−1 + αi ))h¯ i (k) + αi (αi−1 + αi )h¯ i−1 (k)], i = 1, . . . , n − 1, ¯hn (k + 1) − h¯ n (k) = −τ u¯ n (k) = τ αn (αn−1 + αn )(h¯ n−1 (k) − h¯ n (k)). (16) Let h¯ (k) = [h¯ 0 (k), . . . , h¯ n (k)]T , we have h¯ (k + 1) = P h¯ (k),
(17)
where P = In+1 − τ L˜ with L˜ given by Eq. (18) in Box II. When 2 0 < τ < 1/(4αM ), P is a stochastic matrix with positive diagonal elements and the graph corresponding to P contains a spanning tree. According to Lemma 3, P is SIA. Then, lim h¯ (k) = lim P k−km h¯ (km ) = 1v T h¯ (km ),
k→∞
k→∞
(19)
where v is a column vector with nonnegative elements. Note that h¯ (km ) > 0, i = 0, 1, . . . , n holds based on Lemma 1. Therefore, d (q (k),q (k)) limk→∞ ρ αi +α i+1 = c for i = 0, 1, . . . , n where c is a positive i
i+1
number.
j̸=i
≥ (1 − τ αi Lii )¯ui (k) − τ αi
j̸=i
Lij sat(¯uj (k)),
Now we are ready to present the main result of this paper in the following theorem.
L. Dou et al. / Automatica 81 (2017) 464–470
α (α + α ) 1 2 1 .. . ··· L˜ = .. . 0
−α1 (α1 + α2 ) .. .
··· .. .
0
−αi (αi−1 + αi ) .. .
.. . αi+1 (αi+1 + αi+2 ) + αi (αi−1 + αi ) .. .
0
···
469
−αi+1 (αi+1 + αi+2 ) .. . −αn (αn−1 + αn )
0
.. . ··· .. .
.
(18)
αn (αn−1 + αn )
Box II.
Fig. 1. (a) Time evolution of the homogeneous mobile sensors’ positions qi (i = 1, . . . , 5) on a line; (b) The roughness function ρ(x) = 5ex for x ∈ [0, 1], and the final positions of the homogeneous sensors in [0, 1] on the x-axis.
Fig. 2. (a) Time evolution of the heterogeneous mobile sensors’ positions qi (i = 1, . . . , 5) on a line; (b) The control inputs ui (i = 1, . . . , 5) of the heterogeneous mobile sensors.
2 Theorem 1. Suppose condition (1) is satisfied. If 0 < τ < 1/(4αM ), with the proposed coverage control laws (7), networked mobile sensors will converge to the optimal set of positions which minimizes the coverage cost function T (q1 , . . . , qn , ρ) with order preservation.
Proof. Given the initial condition (1), it has been proved in Lemma 1 that the execution of the proposed control laws (7) always preserves the mobile sensors’ ordering if the sampling 2 period τ satisfies 0 < τ < 1/(4αM ). It follows from Lemma 4 that dρ (qi (k), qi+1 (k))/(αi + αi+1 ) will converge to a same positive constant for all i = 0, 1, . . . , n as time goes into infinity. The necessary and sufficient conditions in Lemma 2 are thus satisfied in the limit, and the mobile sensors converge to the set of positions corresponding to the minimum coverage cost. 5. Simulations In this section, the effectiveness of the proposed coverage control laws (7) is illustrated by simulating a group of 5 mobile sensors. The starting positions qi (0), i = 1, . . . , 5 are randomly chosen in the interval (0, 1), satisfying the initial condition (1). The roughness of the terrain is nonuniform which is described by ρ(x) = 5ex . First, we consider a group of homogeneous mobile sensors with the same maximum actuation given by 0.2. The sampling period is chosen as τ = 0.4. Fig. 1(a) displays the evolution of the sensors’ positions. Fig. 1(b) shows the roughness function ρ(x) for x ∈ [0, 1] and the final positions of the sensors on the x-axis. It can be seen from Fig. 1(b) that in the final configuration more sensors are positioned where the roughness function ρ is higher. Next, we consider a group of heterogeneous mobile sensors with different maximum actuations given by [0.2, 0.2, 0.4, 0.1, 0.8]. We set the sampling period as τ = 0.3. The evolution of the heterogeneous sensors’ positions is shown in Fig. 2(a). It can be
Fig. 3. Time evolution of functions Ti and T for a heterogeneous mobile sensor network.
observed from Figs. 1(a) and 2(a) that the sensors’ spatial ordering is preserved throughout the evolution of the networks. Fig. 2(b) shows the evolution of the sensors’ control inputs. We introduce the variables Ti = dρ (qi , qi+1 )/(αi + αi+1 ), i = 0, . . . , 5 to verify whether the optimality condition in Lemma 2 is satisfied. Fig. 3 shows the time evolution of Ti , i = 0, . . . , 5 and the coverage cost function T (q1 , . . . , q5 , ρ). It can be seen that the variables Ti finally achieve a consensus. Meanwhile, the coverage cost function T (q1 , . . . , q5 , ρ) reaches the minimum. 6. Conclusion This paper has proposed a distributed control scheme for a network of heterogeneous mobile sensors to optimally position themselves on a line with varying roughness. Each sensor is assumed to have a different actuation limit. We have developed distributed coverage control laws with input constraints which guarantee the minimization of the largest time required for the mobile sensor network to reach any point on the line. Moreover, with the proposed control laws, the ordering of the sensors is preserved and
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L. Dou et al. / Automatica 81 (2017) 464–470
collisions among the sensors are thus avoided. In real-world applications, the varying roughness of the terrain is often unknown by the sensors and it is of practical significance to develop adaptive coverage control laws for mobile sensor networks deployed in unknown environments. It is also worthwhile to extend the current work to mobile sensors with more realistic dynamics such as point mass following Newton’s second law of motion. References Bartolini, N., Calamoneri, T., La Porta, T., & Silvestri, S. (2011). Autonomous deployment of heterogeneous mobile sensors. IEEE Transactions on Mobile Computing, 10, 753–766. Braun, D. J., Petit, F., Huber, F., Haddadin, S., Van Der Smagt, P., Albu-Schaffer, A., & Vijayakumar, S. (2013). Robots driven by compliant actuators: Optimal control under actuation constraints. IEEE Transactions on Robotics, 29, 1085–1101. Breitenmoser, A., Schwager, M., Metzger, J.-C., Siegwart, R., & Rus, D. (2010). Voronoi coverage of non-convex environments with a group of networked robots. In Proceedings of IEEE international conference on robotics and automation, Anchorage, Alaska (pp. 4982–4989). Brezak, M., & Petrovic, I. (2011). Time-optimal trajectory planning along predefined path for mobile robots with velocity and acceleration constraints. In IEEE/ASME international conference on advanced intelligent mechatronics, Budapest, Hungary (pp. 942–947). Caicedo-Nuez, C., & Žefran, M. (2008). A coverage algorithm for a class of nonconvex regions. In Proceedings of the 47th IEEE conference on decision and control, Cancun, Mexico, December (pp. 4244–4249). Choi, J., & Horowitz, R. (2010). Learning coverage control of mobile sensing agents in one-dimensional stochastic environments. IEEE Transactions on Automatic Control, 55, 804–809. Cortés, J., & Bullo, F. (2005). Coordination and geometric optimization via distributed dynamical systems. SIAM Journal on Control and Optimization, 44, 1543–1574. Cortés, J., Martınez, S., & Bullo, F. (2005). Spatially-distributed coverage optimization and control with limited-range interactions. ESAIM: Control, Optimisation and Calculus of Variations, 11, 691–719. Cortés, J., Martınez, S., Karatas, T., & Bullo, F. (2004). Coverage control for mobile sensing networks. IEEE Transactions on Robotics and Automation, 20, 243–255. Davison, P., Leonard, N. E., Olshevsky, A., & Schwemmer, M. (2015). Nonuniform line coverage from noisy scalar measurements. IEEE Transactions on Automatic Control, 60, 1975–1980. Frasca, P., Garin, F., Gerencsér, B., & Hendrickx, J. M. (2015). Optimal onedimensional coverage by unreliable sensors. SIAM Journal on Control and Optimization, 53, 3120–3140. Jin, Z., & Bertozzi, A.L. (2007). Environmental boundary tracking and estimation using multiple autonomous vehicles. In Proceedings of the 46th IEEE conference on decision control, New Orleans, Lu, December (pp. 4918–4923). Kantaros, Y., Thanou, M., & Tzes, A. (2015). Distributed coverage control for concave areas by a heterogeneous robot–swarm with visibility sensing constraints. Automatica, 53, 195–207. Laventall, K., & Cortés, J. (2009). Coverage control by multi-robot networks with limited-range anisotropic sensory. International Journal of Control, 82, 1113–1121. Lekien, F., & Leonard, N. E. (2009). Nonuniform coverage and cartograms. SIAM Journal on Control and Optimization, 48, 351–372. Leonard, N. E., & Olshevsky, A. (2013). Nonuniform coverage control on the line. IEEE Transactions on Automatic Control, 58, 2743–2755. Li, W., & Cassandras, C.G. (2005). Distributed cooperative coverage control of sensor networks. In Proceedings of the 44th IEEE conference on decision and control and 2005 European control conference (pp. 2542–2547). MartíNez, S., & Bullo, F. (2006). Optimal sensor placement and motion coordination for target tracking. Automatica, 42, 661–668. Miah, S., Nguyen, B. X., Bourque, A., & Spinello, D. (2015). Nonuniform coverage control with stochastic intermittent communication. IEEE Transactions on Automatic Control, 60, 1981–1986. Pimenta, L.C., Kumar, V., Mesquita, R.C., & Pereira, G.A. (2008). Sensing and coverage for a network of heterogeneous robots. In Proceedings of the 47th IEEE conference on decision and control, Cancun, Mexico, December (pp. 3947–3952). Poduri, S., & Sukhatme, G.S. (2004). Constrained coverage for mobile sensor networks. In Proceedings of IEEE International conference on robotics and automation, New Orleans, LA (pp. 165–171). Ren, W., & Beard, R. W. (2005). Consensus seeking in multiagent systems under dynamically changing interaction topologies. IEEE Transactions on Automatic Control, 50, 655–661. Song, C., Liu, L., Feng, G., Wang, Y., & Gao, Q. (2013). Persistent awareness coverage control for mobile sensor networks. Automatica, 49, 1867–1873. Song, C., Liu, L., Feng, G., & Xu, S. (2016). Coverage control for heterogeneous mobile sensor networks on a circle. 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Liya Dou received the B.S. degree in information and computing science, and M.S. degree in mathematics from University of Science and Technology Beijing, Beijing, China, in 2011 and 2014, respectively. She is now working towards her joint Ph.D. degree with the Department of Mechanical and Biomedical Engineering, City University of Hong Kong, Hong Kong SAR, China, and also with the Department of Automation, Shanghai Jiao Tong University, Shanghai, China. Her research interests include cooperative control and multi-agent networks.
Cheng Song received the B.Eng. degree in Automation and the Ph.D. degree in Control Science and Engineering from the University of Science and Technology of China, Hefei, China, in 2007 and 2012, respectively, and the Ph.D. degree in the Department of Mechanical and Biomedical Engineering from the City University of Hong Kong, Kowloon, Hong Kong, in 2012. He is currently an Associate Professor in the School of Automation, Nanjing University of Science and Technology. His main research interests include cooperative control, multi-agent systems, and mobile sensor networks.
Xiaofan Wang received the Ph.D, degree from Southeast University, China in 1996. He has been a Professor in the Department of Automation, Shanghai Jiao Tong University (SJTU) since 2002 and a Distinguished Professor of SJTU since 2008. He received the 2002 National Science Foundation for Distinguished Young Scholars of P. R. China, the 2005 Guillemin-Cauer Best Transactions Paper Award from the IEEE Circuits and Systems Society, the 2008 Distinguished Professor of the Chang Jiang Scholars Program, Ministry of Education, and the 2015 Second Class Prize of the State Natural Science Award. His current research interests include analysis and control of complex dynamical networks. He is currently the Chair of the IFAC Technical Committee on Large-Scale Complex Systems, Board member of the international Network Science Society (NetSci) and Chair of the Chinese Technical Committee on Complex Networks an System Control.
Lu Liu received her Ph.D. degree in 2008 in the Department of Mechanical and Automation Engineering, Chinese University of Hong Kong, Hong Kong. From 2009 to 2012, she was an Assistant Professor in University of Tokyo, Japan, and then a Lecturer in University of Nottingham, United Kingdom. She is currently an Assistant Professor in the Department of Mechanical and Biomedical Engineering, City University of Hong Kong, Hong Kong. Her research interests are primarily in networked dynamical systems, control theory and applications, and biomedical devices. She received the Best Paper Award (Guan Zhaozhi Award) in the 27th Chinese Control Conference in 2008.
Gang Feng received the B.Eng. and M.Eng. degrees in Automatic Control from Nanjing Aeronautical Institute, China in 1982 and in 1984 respectively, and the Ph.D. degree in Electrical Engineering from the University of Melbourne, Australia in 1992. He has been with City University of Hong Kong since 2000 where he is now Chair Professor of Mechatronic Engineering. He was lecturer/senior lecturer at School of Electrical Engineering, University of New South Wales, Australia, 1992–1999. He was awarded an Alexander von Humboldt Fellowship in 1997, and the IEEE Transactions on Fuzzy Systems Outstanding Paper Award in 2007, and Changjiang chair professorship from Education Ministry of China in 2009. His current research interests include multi-agent systems and control, intelligent systems and control, and networked systems and control. Prof. Feng is an IEEE Fellow, an associate editor of IEEE Trans. Fuzzy Systems and Journal of Systems Science and Complexity, and was an associate editor of IEEE Trans. Automatic Control, IEEE Trans. Systems, Man & Cybernetics, Part C, Mechatronics, and Journal of Control Theory and Applications.
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Automatica journal homepage: www.elsevier.com/locate/automatica
Brief paper
Nonuniform coverage control for heterogeneous mobile sensor networks on the line✩ Liya Dou a,b , Cheng Song c , Xiaofan Wang a , Lu Liu b,1 , Gang Feng b a
Department of Automation, Shanghai Jiao Tong University, Shanghai 200240, China
b
Department of Mechanical and Biomedical Engineering, City University of Hong Kong, Kowloon, Hong Kong Special Administrative Region
c
School of Automation, Nanjing University of Science and Technology, Nanjing 210094, China
article
info
Article history: Received 16 May 2016 Received in revised form 21 February 2017 Accepted 14 March 2017
Keywords: Nonuniform coverage Mobile sensor network Distributed control Actuation constraint
abstract The coverage control problem for networked mobile sensors on a line with different actuation limits is addressed in this paper. The roughness of each point on the line is assumed to be different which makes the concerned problem more challenging. The objective of coverage control considered in this paper is to minimize the largest time required for the sensor network to reach any point on the line via optimizing the sensors’ locations on this line. Distributed coverage control laws with input constraints are developed to drive the sensors to the optimal configuration while preserving their spatial ordering on the line. Simulation examples demonstrate the effectiveness of the proposed control laws. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction Networks of sensors and autonomous vehicles have wide applications in many fields, including environment monitoring, surveillance, industrial diagnostics and so on. When the mission domain is remote and/or hostile, such networks must be capable of operating in a fully autonomous and distributed way to adapt to changing environments. The potential of such applications has prompted much interest in coverage control of mobile sensor networks, where the goal is to drive networked mobile sensors to an optimal sensing configuration in a mission field. If the information density or terrain roughness is uniform over the mission field, the problem is referred to as the uniform coverage control problem, whereas for a nonuniform field, the problem is referred to as the nonuniform coverage control problem with a nonuniform metric.
✩ This work was supported by grants from the Research Grants Council of Hong Kong (Nos. CityU-11213415, CityU-11261516), the National Natural Science Foundation of China (Nos. 61374176, 61403203), and the Natural Science Foundation of Jiangsu Province (No. BK20140798). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Michael M. Zavlanos under the direction of Editor Christos G. Cassandras. E-mail addresses: [email protected] (L. Dou), [email protected] (C. Song), [email protected] (X. Wang), [email protected] (L. Liu), [email protected] (G. Feng). 1 Fax: +852 3442 0172.
http://dx.doi.org/10.1016/j.automatica.2017.04.029 0005-1098/© 2017 Elsevier Ltd. All rights reserved.
In practice, mobile sensors may have some physical constraints like limited communication or sensing ranges (Laventall & Cortés, 2009), velocity or actuation constraints (Braun et al., 2013; Brezak & Petrovic, 2011). Accordingly, a network of heterogeneous mobile sensors in terms of their different actuation limits is considered in this work. This paper investigates the nonuniform coverage control of a network of heterogeneous mobile sensors located on a line with varying roughness. 1.1. Related work There is a large volume of works on coverage control problems of sensor networks in the open literature, see for examples Bartolini, Calamoneri, La Porta, and Silvestri (2011), Breitenmoser, Schwager, Metzger, Siegwart, and Rus (2010), Caicedo-Nuez and Žefran (2008), Cortés and Bullo (2005), Cortés, Martınez, and Bullo (2005); Cortés, Martınez, Karatas, and Bullo (2004), Kantaros, Thanou, and Tzes (2015), Laventall and Cortés (2009), Li and Cassandras (2005), MartíNez and Bullo (2006), Pimenta, Kumar, Mesquita, and Pereira (2008), Poduri and Sukhatme (2004), Song, Liu, Feng, Wang, and Gao (2013), Zhai and Hong (2013) and references therein. Based on Voronoi partition and gradient descent algorithms, coverage control laws are developed for a group of mobile sensors in uniform fields in Cortés and Bullo (2005). Uniform coverage control with a constraint on the minimum number of each agent’s neighbors is investigated in Poduri and Sukhatme (2004).
L. Dou et al. / Automatica 81 (2017) 464–470
Nonuniform coverage control of sensor networks is addressed in part by using density-dependent gradient descent laws in Cortés et al. (2004), Li and Cassandras (2005) and Pimenta et al. (2008). Nonuniform coverage control of mobile agents with limited sensing or communication range is treated in Cortés et al. (2005), Kantaros et al. (2015) and Laventall and Cortés (2009). The authors in Lekien and Leonard (2009) investigate the nonuniform coverage problem of a planar region with a non-Euclidean distance metric that stretches and shrinks regions of high and low density, respectively. Nonuniform coverage control laws are proposed using cartogram transformation which requires some global information about the region. In addition, nonuniform coverage of a planar region by a group of mobile sensors with stochastically intermittent communications among them is studied in Miah, Nguyen, Bourque, and Spinello (2015). 1.2. One-dimensional coverage problem This paper focuses on the nonuniform coverage problem when the sensors are arranged on a line, which is motivated by two considerations. First, the one-dimensional coverage problem has several potential applications such as environmental boundary monitoring and target tracking (Jin & Bertozzi, 2007; Susca, Bullo, & Martínez, 2008), and it has attracted increasing attention in recent years, such as Choi and Horowitz (2010), Davison, Leonard, Olshevsky, and Schwemmer (2015), Frasca, Garin, Gerencsér, and Hendrickx (2015). When a group of mobile sensors performs border patrol, they should be optimally positioned on a curve. This problem can be transformed to the line coverage problem investigated in this paper by parametrizing the curve with its arc-length. Second, the one-dimensional coverage problem can provide a simplified setting for addressing unsettled issues in coverage control. One typical issue is to design control algorithms for mobile agents to minimize the response time or energy cost from the agents to any point in the mission field. The paper Leonard and Olshevsky (2013) develops a distributed coverage control law for a nonuniform field under which a group of mobile agents are optimally positioned on a line, such that the distance from the mobile agents to any point on the line is minimized. Taking into consideration the sensors’ velocity constraints, the authors in Song, Liu, Feng, and Xu (2016) study the uniform coverage problem of a circle by networked mobile sensors to minimize the largest time taken from the network to any point on the circle. 1.3. Our contributions In this paper, we consider the nonuniform coverage control of a network of heterogeneous mobile sensors with different actuation limits located on a line. The roughness of each point on the line is assumed to be different. Our objective is to minimize a coverage cost function, defined as the largest time required for the sensor network to reach any point on the line. Distributed control laws are designed, under which the sensors are driven to the optimal positions with order preservation. Compared with Leonard and Olshevsky (2013), where nonuniform coverage control laws are designed for homogeneous sensors without actuation constraints, this paper takes into account different actuation limits of mobile sensors and varying roughness of the terrain, and designs distributed control laws with input saturation. The problem investigated in this paper is more general. Compared with Song et al. (2016), where coverage control problem of networked mobile sensors with different velocity constraints positioned on a circle is studied, this paper investigates the coverage problem of mobile sensors with actuation constraints located on a line, and also takes varying roughness of the terrain into consideration.
465
The remainder of this paper is structured as follows. Section 2 formalizes the nonuniform coverage problem with actuation constraints. Distributed coverage control laws are proposed in Section 3. Section 4 presents the convergence analysis of the coverage control laws. In Section 5, the effectiveness of the results is illustrated by simulations. Finally, Section 6 concludes the paper. 2. Problem formulation Consider a network of n mobile sensors initially located at arbitrary positions q1 (0), q2 (0), . . . , qn (0) which are assumed, without loss of generality, to lie in the interval [0, 1]. Denote R+ as the set of positive real numbers. Let ρ : [0, 1] → R+ , be a positive, piecewise-continuous function, which measures the roughness at each point on the line. We assume that ρ is bounded, i.e., there exist positive constants ρmax and ρmin such that for all z ∈ [0, 1], we have ρmin ≤ ρ(z ) ≤ ρmax . In this work, the roughness function ρ is assumed to be known a priori by the sensors. Following Leonard and Olshevsky (2013), the distance between a and b ∈ [0, 1] is defined as dρ (a, b) =
max(a,b) min(a,b)
ρ(z )dz ,
b ρ(z )dz. It is easy to see that and we denote d¯ ρ (a, b) = a dρ (a, b) = |d¯ ρ (a, b)|. We label the sensors from 1 to n in accordance with their initial order along the line from left to right and assume that no two sensors occupy the same position, that is,
0 < q1 (0) < q2 (0) < · · · < qn (0) < 1.
(1)
We introduce the function F (x) =
x
ρ(z )dz . 0
Note that F (1) = dρ (0, 1). Furthermore, for any two points a < b in [0, 1], dρ (a, b) = F (b) − F (a). In this paper, we consider the nonuniform coverage problem for heterogeneous mobile sensors with actuation constraints. There exists a maximum actuation αi for each sensor. Networked mobile sensors are assumed to evolve in discrete time periods. The maximum distance that each sensor can move in one sampling period is limited, which depends on both the sensor’s actuation limit and the varying roughness of the terrain. The actuation needed q for each agent to move from point p to q on the line is given by | p f (ρ(z ))dz |, where f (x) : R+ → R+ is a strictly increasing continuous function with respect to x. This definition captures the fact that the agents need larger actuation for movement in areas where the roughness function ρ is higher, and smaller actuation in areas where ρ is lower. For simplicity, we assume f (ρ(z )) = ρ(z ) in this work. As sensor i has the maximum actuation αi , the actuation for sensor i to move on the line in one period is no larger
q (k+1)
than αi , that is, −τ αi ≤ q i(k) ρ(z )dz ≤ τ αi for i = 1, . . . , n, i and k ≥ 0, where qi (k) is the position of sensor i on the line at the discrete-time index k and τ > 0 is the sampling period. According to the above considerations, the evolution of the mobile sensors’ positions depends on not only the control inputs but also the varying roughness of the terrain. For our analysis, each mobile sensor is assumed to evolve according to
qi (k+1) qi (k)
ρ(z )dz = τ ui (k),
(2)
where ui (k) is the control input of sensor i at the discrete-time index k. Since τ > 0 is the sampling period, ui (k) also can be regarded
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as the actuation of sensor i moving on the line with varying roughness at the discrete-time index k. Due to the fact that the sensors have different actuation limits, it is reasonable to consider control inputs with different constraints, that is, −αi ≤ ui (k) ≤ αi , k ≥ 0, i = 1, . . . , n. In order to make the networked mobile sensors’ dynamics easy to analyze, we define yi (k) = F (qi (k)), i = 1, . . . , n. From (2), we can conclude that the variables yi (k) evolve as follows, yi (k + 1) = yi (k) + τ ui (k),
i = 1 , . . . , n.
(3)
We assume that there exists an upper bound αM on the sensors’ actuation limits and it is known by the networked mobile sensors beforehand. To simplify notation, let q0 (k) = 0, qn+1 (k) = 1, y0 (k) = 0 and yn+1 (k) = F (1) for all k, and α0 = αn+1 = 0 throughout the paper. We define the coverage cost function as T (q1 , . . . , qn , ρ) = max min
q∈[0,1] i=1,...,n
dρ (qi , q)
αi
,
where dρ (qi , q) is the weighted distance between sensor i and point q on a line with varying roughness. The time required for mobile sensor i to move to point q on the line depends on the weighted distance dρ (qi , q) and the actuation limit of sensor i. From (2), we know that αi denotes the maximum weighted distance that sensor i can move in one sampling period. Therefore, dρ (qi , q)/αi indicates the shortest time required for sensor i to arrive at point q. Define the time required for a mobile sensor network to reach a point on the line as the minimum of the shortest time required for the sensors to arrive at the point. As a result, T represents the largest time required for the network to reach any point on the line. Our objective is to drive networked mobile sensors to the optimal coverage configuration, that is, arg minq1 ,...,qn T (q1 , . . . , qn , ρ), by designing distributed control laws. 3. Distributed coverage control laws
i = 1 , . . . , n,
2 Lemma 1. Suppose condition (1) is satisfied. If 0 < τ < 1/(4αM ), then with the coverage control laws (7), 0 < q1 (k) < q2 (k) < · · · < qn (k) < 1 holds for all k ≥ 0.
Proof. First we prove that the ordering of the sensors is preserved by using the following control inputs without saturation u˜ i (k) = αi u¯ i (k),
u¯ 1 (k) = α1 d¯ ρ (q1 (k), q2 (k)) − (α1 + α2 )d¯ ρ (0, q1 (k)), u¯ i (k) = (αi−1 + αi )d¯ ρ (qi (k), qi+1 (k)) − (αi + αi+1 ) ×d¯ ρ (qi−1 (k), qi (k)), i = 2, . . . , n − 1, ¯un (k) = (αn−1 + αn )d¯ ρ (qn (k), 1) − αn d¯ ρ (qn−1 (k), qn (k)).
(5)
yi (k + 1) = yi (k) + τ αi u¯ i (k),
i = 1, . . . , n.
Define h0 (k) = y1 (k), hi (k) = yi+1 (k) − yi (k), hn (k) = F (1) − yn (k).
i = 1, . . . , n − 1,
The control laws u¯ i , i = 1, . . . , n can be rewritten in the following form u¯ 1 (k) = α1 h1 (k) − (α1 + α2 )h0 (k), u¯ i (k) = (αi−1 + αi )hi (k) − (αi + αi+1 )hi−1 (k),
i = 2, . . . , n − 1, (11)
u¯ n (k) = (αn−1 + αn )hn (k) − αn hn−1 (k).
Recall that α0 = αn+1 = 0, y0 (k) = 0 and yn+1 (k) = F (1). Therefore, under the control laws (10), if hi (k) > 0, and hi−1 (k) > 0, i = 1, . . . , n, one has
= τ αi [(αi−1 + αi )hi (k) − (αi + αi+1 )hi−1 (k)] ≤ 2τ αM2 hi (k), i = 1, . . . , n. Similarly, i = 0, . . . , n − 1.
The above two inequalities imply that yi+1 (k + 1) − yi+1 (k) − (yi (k + 1) − yi (k))
(6)
We can rewrite compactly the control laws (4) as u(k) = α sat(¯u(k)),
(10)
2 yi+1 (k + 1) − yi+1 (k) ≥ −2τ αM hi (k),
It is noted that for i = 2, . . . , n − 1, u¯ i makes use of the weighted distance over the roughness ρ from sensor i to sensors i − 1 and i + 1, that is, d¯ ρ (qi−1 (k), qi (k)) and d¯ ρ (qi (k), qi+1 (k)), respectively. It means that sensor i coordinates its motion with that of sensors i − 1 and i + 1 for i = 2, . . . , n − 1. Particularly, sensor 1 updates its position based on the distance from it to the border 0 and sensor 2, and sensor n updates its position in a similar way. By the definition of yi (k), Eq. (5) can be equivalently written as u¯ 1 (k) = α1 (y2 (k) − y1 (k)) − (α1 + α2 )(y1 (k) − 0), u¯ i (k) = (αi−1 + αi )(yi+1 (k) − yi (k)) − (αi + αi+1 ) ×(yi (k) − yi−1 (k)), i = 2, . . . , n − 1, u¯ n (k) = (αn−1 + αn )(F (1) − yn (k)) − αn (yn (k) − yn−1 (k)).
i = 1 , . . . , n,
where u¯ i (k) is given by Eq. (6). In this case, the mobile sensors evolve as follows,
(4)
where sat(·) : R → [−1, 1] is a standard saturation function defined as sat(x) = sign(x) min{1, |x|}, and
(7)
where u(k) = [u1 (k), u2 (k), . . . , un (k)] , α is a diagonal matrix with the ith diagonal element being αi , and u¯ (k) = [¯u1 (k), u¯ 2 (k), . . . , u¯ n (k)]T is given by T
u¯ (k) = −Ly(k) + B
Collisions can be avoided during the coverage task if the ordering of the mobile sensors is preserved. The following lemma shows that the execution of the coverage control laws (7) preserves the sensors’ ordering.
yi (k + 1) − yi (k) = τ αi u¯ i (k)
We propose the coverage control laws as follows ui (k) = αi sat(¯ui (k)),
with y(k) = [y1 (k), y2 (k), . . . , yn (k)]T , B = [0, . . . , 0, (αn−1 + αn )F (1)]T and L is given by Eq. (9) in Box I.
(8)
≥ −4τ αM2 hi (k),
i = 0, 1, . . . , n,
that is 2 hi (k + 1) ≥ (1 − 4τ αM )hi (k),
i = 0, 1, . . . , n,
2 which is larger than zero provided that 0 < τ < 1/(4αM ). Consequently, yi+1 (k + 1) > yi (k + 1) if yi+1 (k) > yi (k) for i = 0, 1, . . . , n. Since F (x) is a strictly monotonically increasing function, it is equivalent to say that qi+1 (k + 1) > qi (k + 1) if qi+1 (k) > qi (k) for i = 0, 1, . . . , n. Thus, with the control laws (10) the ordering of the sensors is preserved. Then we show that the coverage control laws (7) with saturation also preserve the ordering of the sensors. With the control laws (7), we can obtain
h0 (k + 1) = h0 (k) + τ α1 sat(¯u1 (k)), hi (k + 1) = hi (k) + τ (αi+1 sat(¯ui+1 (k)) − αi
× sat(¯ui (k))), i = 1, . . . , n − 1, hn (k + 1) = hn (k) − τ αn sat(¯un (k)).
(12)
L. Dou et al. / Automatica 81 (2017) 464–470
2α + α 1 2 .. . L = ··· .. . 0
−α1 .. .
··· .. .
0
467
0
− (αi + αi+1 ) .. .
.. . αi−1 + 2αi + αi+1 .. .
− (αi−1 + αi ) .. .
0
···
−αn
.. . ··· .. .
.
(9)
αn−1 + 2αn
Box I.
We will prove that given the sampling period 0 < τ < 2 1/(4αM ), hi (k+1) > 0 holds if hi (k) > 0 for i = 1, . . . , n−1. When u¯ i+1 (k) ≥ 1 and u¯ i (k) < −1, hi (k + 1) = hi (k) + τ (αi+1 + αi ) > 0 if hi (k) > 0. When u¯ i+1 (k) ≥ 1 and −1 ≤ u¯ i (k) ≤ 1, hi (k + 1) = hi (k) + τ (αi+1 − αi u¯ i (k))
= τ αi+1 + (1 − τ αi (αi−1 + αi ))hi (k) + τ αi (αi + αi+1 )hi−1 (k) > 0 2 if hi−1 (k) > 0, hi (k) > 0 and 0 < τ < 1/(4αM ). When u¯ i+1 (k) ≥ 1 and u¯ i (k) > 1, hi (k + 1) = hi (k) + τ (αi+1 − αi ). Since u¯ i (k) = (αi−1 + αi )hi (k) − (αi + αi+1 )hi−1 (k) ≥ 1 and hi−1 (k) > 0, we have that hi (k) > 1/(αi−1 + αi ). Therefore, hi (k + 1) > 1/(αi−1 + αi ) + τ (αi+1 − αi ) > 0 holds when 2 0 < τ < 1/(4αM ). When u¯ i+1 (k) ≤ −1 and u¯ i (k) ≤ −1, hi (k + 1) = hi (k) + τ (αi − αi+1 ). Similarly, it can be shown that in 2 this case hi (k + 1) > 0 if hi+1 (k) > 0 and 0 < τ < 1/(4αM ). When −1 ≤ u¯ i+1 (k) ≤ 1 and u¯ i (k) ≤ −1,
hi (k + 1) = hi (k) + τ (αi+1 u¯ i+1 (k) + αi )
= τ αi + (1 − τ αi+1 (αi+1 + αi+2 ))hi (k) + τ αi+1 (αi + αi+1 )hi+1 (k) > 0 2 if 0 < τ < 1/(4αM ), hi (k) > 0 and hi+1 (k) > 0. When u¯ i+1 (k) < 1 and u¯ i (k) ≥ −1, we can obtain that hi (k + 1) ≥ hi (k) + τ (αi+1 u¯ i+1 (k) − αi u¯ i (k)) > 0 holds since it has been proved that hi (k + 1) = hi (k) + τ (αi+1 u¯ i+1 (k) − αi u¯ i (k)) > 0 if hi (k) > 0 for i = 1, . . . , n − 1 under the coverage control laws (10). For h0 (k), we note that h0 (k + 1) > h0 (k) + τ α1 u¯ 1 (k) when u¯ 1 (k) < 1. As we have shown that h0 (k + 1) = h0 (k) + τ α1 u¯ 1 (k) > 0 if h0 (k) > 0, h0 (k + 1) > 0 holds if h0 (k) > 0 when u¯ 1 (k) < 1. When u¯ 1 (k) ≥ 1, it is easy to see that h0 (k + 1) = h0 (k)+τ α1 > 0 if h0 (k) > 0. Similarly, we can prove that hn (k + 1) > 0 if hn (k) > 0. In conclusion, hi (k + 1) > 0 if hi (k) > 0 for i = 0, 1, . . . , n with the control laws (7). If condition (1) is satisfied and 0 < τ < 2 1/(4αM ), with the proposed coverage control laws (7), 0 < q1 (k) < · · · < qi (k) < qi+1 (k) < · · · < qn (k) < 1 holds for all k ≥ 0.
Remark 1. Under the given conditions in Lemma 1, 0 < q1 (k) < q2 (k) < · · · < qn (k) < 1 holds for all k ≥ 0. In this case, d¯ ρ (qi (k), qi+1 (k)) = dρ (qi (k), qi+1 (k)) for i = 0, . . . , n, and k ≥ 0. From (5), one has −2αM dρ (qi−1 (k), qi (k)) < u¯ i (k) < 2αM dρ (qi (k), qi+1 (k)). Since ui (k) = αi sat(¯ui (k)), ui (k) ≤ αi u¯ i (k) when u¯ i (k) ≥ 0, and ui (k) ≥ αi u¯ i (k) when u¯ i (k) < 0. Thus, −2αM2 dρ (qi−1 (k), qi (k)) < ui (k) < 2αM2 dρ (qi (k), qi+1 (k)). Since 2 0 < τ < 1/(4αM ) and ρ(z ) > 0, according to Eq. (2), we have
qi (k)
qi (k+1)
qi+1 (k)
ρ(z )dz < qi (k) ρ(z )dz < ρ(z )dz, and qi (k) consequently qi−1 (k) < qi (k + 1) < qi+1 (k) for i = 1, . . . , n. that −
1 2
qi−1 (k)
1 2
Note also that each mobile sensor’s neighbors are fixed since the execution of the control laws (7) preserves the sensors’ ordering. When implementing the control laws (7), sensor i, i = 2, . . . , n−1 communicates with its left neighbor and right neighbor to acquire the information of their positions and actuation limits. Then it will compute the weighted distance dρ (qi−1 (k), qi (k)) to its left neighbor and dρ (qi (k), qi+1 (k)) to its right neighbor. Based on all the information, sensor i can compute the control law ui (k), and
moves to the position in the interval (qi−1 (k), qi+1 (k)) satisfying Eq. (2). Sensors 1 and n are border sensors with a single neighbor. Sensor 1 computes its control law based on the weighted distance to its right neighbor and the left boundary of the line. Sensor n can implement the control law un (k) in a similar way. Therefore, the proposed coverage control laws (7) can be executed in a distributed way without knowledge of the sensors’ labels. 4. Convergence analysis In this section, we show that with the proposed control laws (7) a heterogeneous mobile sensor network will converge to the optimal set of positions which corresponds to the minimum coverage cost. We first give necessary and sufficient conditions for achieving the minimum of the function T (q1 , . . . , qn , ρ). Lemma 2. The coverage cost function T (q1 , . . . , qn , ρ) is minimized if and only if dρ (0, q1 )
α1
=
dρ (q1 , q2 )
= ··· =
α1 + α2
dρ (qn−1 , qn )
αn−1 + αn
=
dρ (qn , 1)
αn
. (13)
Moreover, the minimum of the function T (q1 , . . . , qn , ρ) is F (1)/ (2 nl=1 αl ). Proof. Define ρ
Vi =
q ∈ [0, 1]|
dρ (qi , q)
≤
αi
dρ (qj , q)
αj
, ∀j ̸= i .
(14)
x
ρ(z )dz is monotonically increasAs the function F (x) = 0 ρ ing, each set Vi , i = 1, . . . , n is a closed set.We can also ρ = [0, 1]. Therefore, ni=1 V ρ ρ(z )dz ≥ have that i=1,...,n Vi i
F (1). From the definition of the function T (q1 , . . . , qn , ρ) and ρ the sets Vi , i = 1, . . . , n, we have T (q1 , . . . , qn , ρ) = dρ (qi ,q)
d ( q ,q )
= maxi=1,...,n maxq∈V ρ ρ αii . Note i n ρ ρ that F (1) ≤ i=1 V ρ(z )dz ≤ 2 i=1 maxq∈Vi dρ (qi , q) ≤ i n n 2T i=1 αi . Therefore, T (q1 , . . . , qn , ρ) ≥ F (1)/(2 l=1 αl ) almaxq∈[0,1] mini=1,...,n
n
αi
ways holds. (Sufficiency) Recall that q0 (k) = 0, qn+1 (k) = 1, and α0 = αn+1 = 0. From the definition of the function T (q1 , . . . , qn , ρ), we have T (q1 , . . . , qn , ρ) = max min
dρ (qi , q)
αi
q∈[0,1] i=1,...,n
= max
max
min
dρ (qi , q)
αi
i=0,...,n q∈[qi ,qi+1 ] i=1,...,n
≤ max
max
dρ (q, q1 )
q∈[q0 ,q1 ]
α1
max
max min i=1,...,n−1 q∈[qi ,qi+1 ] max
q∈[qn ,qn+1 ]
dρ (qn , q)
αn
, dρ (qi , q) dρ (q, qi+1 )
αi
.
,
αi+1
,
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L. Dou et al. / Automatica 81 (2017) 464–470
d q,q ρ , ρ (αi+1i+1 ) , i = 1, . . . , n − 1, d0 = αi dρ (q,q1 ) dρ (qn ,q) ρ , and dn = . For an arbitrary point located in αi αn ρ [qi , qi+1 ], we have (αi + αi+1 )di ≤ dρ (qi , q) + dρ (q, qi+1 ) = ρ dρ (qi , qi+1 ), i = 1, . . . , n − 1, α1 d0 = dρ (q, q1 ) ≤ dρ (q0 , q1 ), ρ ρ and αn dn = dρ (qn , q) ≤ dρ (qn , qn+1 ). It implies that di ≤
ρ
Denote di = min
dρ (qi ,q)
dρ (qi , qi+1 )/(αi + αi+1 ), i = 0, . . . , n holds for all points located in [qi , qi+1 ]. Therefore, T (q1 , . . . , qn , ρ) ≤ maxi=0,...,n ρ maxq∈[qi ,qi+1 ] di ≤ maxi=0,...,n dρ (qi , qi+1 )/(αi + αi+1 ). When dρ (qi , qi+1 )/(αi + αi+1 ) is identical for i = 0, . . . , n, we can obn tain dρ (qi , qi+1 )/(αi + αi+1 ) = F (1)/(2 l=1 αl ), i = 0, . . . , n, n = F (1). As we have shown that by the fact i=0 dρ (qi , qi+1 ) n T (q1 , . . . , qn , ρ) ≥ F (1)/(2 l=1 αl ) holds, T (q1 , . . . , qn , ρ) = n F (1)/(2 l=1 αl ) when dρ (qi , qi+1 )/(αi + αi+1 ) is identical. (Necessity) If the function T (q1 , . . . , qn , ρ) is minimized, dρ (qi ,q)
T (q1 , . . . , qn , ρ) = maxi=1,...,n maxq∈V ρ i
αi
= F (1)/(2
n
l =1
d (q ,q) αl ). Then, maxq∈V ρ ρ αii ≤ F (1)/(2 nl=1 αl ). In consequence, i n ρ ρ(z )dz ≤ 2 maxq∈V ρ dρ (qi , q) ≤ F (1)αi / l=1 αl holds for V i
i
ρ
i = 1, . . . , n. If there exists a set Vj such that maxq∈V ρ j n n ρ ρ(z )dz < F (1)/(2 l=1 αl ), we have that i =1 V contradicts with the fact
n i =1
maxq∈V ρ dρ (qi , q) = F (1)αi /(2 i
dρ (qj ,q)
αi
F (1). It
i
ρ
Vni
<
ρ(z )dz ≥ F (1). Therefore,
αl ) for i = 1, . . . , n. Define qi qr Wi = [ , ], i = 1, . . . , n with l ρ(z )dz = q i ρ(z )dz = q i qli
l=1
qri
i
ρ
maxq∈V ρ dρ (qi , q), and we have that Vi ⊆ Wi . Combining with the i
ρ = [0, 1], it can be obtained that [0, 1] ⊆ i=1,...,n ρ(z )dz ≥ F (1). Note that maxq∈V ρ dρ (qi , q) = i ( )α /( α )holds for i = 1, . . . , n. From the definition of n Wi , we have that ρ( z ) dz = F ( 1 ) . Thus, i=1 Wi i=1,...,n Wi = [0, 1] and the interiors of the n closed sets W are disjoint. Recall i ρ ρ that the sets Vi , i = 1, . . . , n are closed sets, i=1,...,n Vi = [0, 1] ρ ρ and Vi ⊆ Wi . Therefore, Vi = Wi for all i = 1, . . . , n. As a n result, dρ (qi , qi+1 )/(αi + αi+1 ) = F (1)/(2 l=1 αl ) holds for all i = 0, . . . , n.
fact
i=1,...,n Vi n Wi and i = 1 Wi n F 1 i 2 l =1 l
Before we show that the optimality condition proposed in Lemma 2 can be satisfied as time goes to infinity, the following lemma will be introduced. Lemma 3 (Ren & Beard, 2005). Let A be a stochastic matrix with positive diagonal elements. If the graph corresponding to A contains a spanning tree, then A is indecomposable and aperiodic (SIA), that is, limk→∞ Ak = 1v T , where v satisfies AT v = v and 1T v = 1. Furthermore, each element of v is nonnegative. 2 Lemma 4. Suppose condition (1) is satisfied. If 0 < τ < 1/(4αM ),
with the control laws (7),
dρ (qi (k),qi+1 (k))
αi +αi+1
, i = 0, 1, . . . , n will con-
verge to the same limit. Proof. From the definition of u¯ (k), we have u¯ (k + 1) − u¯ (k) = −L(y(k + 1) − y(k)) = −τ α Lsat(¯u(k)).
(15)
First it can be proved that for any i = 1, . . . , n, u¯ i (k + 1) ≥ −1 if u¯ i (k) ≥ −1. Note that when u¯ i (k) ≥ −1, sat(¯ui (k)) ≤ u¯ i (k). Then we have
u¯ i (k + 1) = u¯ i (k) − τ αi Lii sat(¯ui (k)) +
Lij sat(¯uj (k))
where Lij denotes the (i, j) term of the matrix L. As Lij ≤ 0, ∀j ̸= i 2 and 1 − τ αi Lii > 0 for i = 1, . . . , n, when 0 < τ < 1/(4αM ), we n ¯ have that ui (k + 1) ≥ −1 + τ αi Lii + τ αi j̸=i Lij . Since j=1 Lij ≥ 0 for i = 1, . . . , n, u¯ i (k + 1) ≥ −1 holds. Similarly, we can prove that u¯ i (k + 1) ≤ 1 if u¯ i (k) ≤ 1. Then we will prove that u¯ (k) will in finite time enter the set Ω = {¯u ∈ Rn : |¯ui (k)| ≤ 1, i = 1, . . . , n}. For any i = 1, . . . , n, |¯ui (k + 1)| ≤ 1 if |¯ui (k)| ≤ 1. Therefore, if |¯ui (0)| ≤ 1 for all i = 1, . . . , n, |¯ui (k)| ≤ 1, ∀k > 0 holds for all i = 1, . . . , n. Ω is an invariant set of system (15). We will prove by contradiction that in finite time u¯ i (k), i = 1, . . . , n will be less than 1. Assume that there are p elements of u¯ (k) larger than 1 for all k ≥ 0, that is, u¯ l (k) > 1, l ∈ {1, . . . , n}, l = l1 , . . . , lp for k ≥ 0. According to the mobile sensors’ dynamics (3), we have that yl (k + 1) = yl (k) + τ αl ,
l = l1 , . . . , lp .
Then, yl (k) = yl (0) + τ αl k, l = l1 , . . . , lp for k ≥ 0. Since τ and αl are both positive constants, we can conclude that yl (k) → ∞ as k → ∞. It contradicts with the fact yl (k) < F (1), ∀l ∈ {1, . . . , n} from Lemma 1. Therefore, u¯ l (k), l = l1 , . . . , lp will be less than 1 at some finite discrete-time index kl . As we have shown that for any i = 1, . . . , n, u¯ i (k + 1) ≥ −1 if u¯ i (k) ≥ −1, u¯ l (k), l = l1 , . . . , lp will enter the interval [−1, 1] at the discrete-time index kl . Upon setting ks = max{kl1 , . . . , klp }, we have that all u¯ l (k), ∀l = l1 , . . . , lp will enter the interval [−1, 1] at the finite discrete-time index ks . Similarly, we can show that in finite time all the elements of u¯ which are less than −1 will enter the interval [−1, 1]. Thus, u¯ (k) will in finite time enter the invariant set Ω . 2 Finally, we will prove that if 0 < τ < 1/(4αM ) and u¯ (k) belongs h (k)
i = c, to the set Ω at a finite discrete-time index km , limk→∞ α +α i i+1 i = 0, 1, . . . , n where c is a positive number. Note that u¯ (k) ∈ Ω holds for all k ≥ km . Then we have u(k) = α u¯ (k), k ≥ km . Denote hi (k) h¯ i (k) = α +α for i = 0, 1, . . . , n. From Eqs. (11) and (12), it can i
i+1
be derived that h¯ 0 (k + 1) − h¯ 0 (k) = τ u¯ 1 (k)
= τ α1 (α1 + α2 )(h¯ 1 (k) − h¯ 0 (k)), τ (αi+1 u¯ i+1 (k) − αi u¯ i (k)) h¯ i (k + 1) − h¯ i (k) = αi + αi+1 = τ [αi+1 (αi+1 + αi+2 )h¯ i+1 (k) − (αi+1 (αi+1 + αi+2 ) + αi (αi−1 + αi ))h¯ i (k) + αi (αi−1 + αi )h¯ i−1 (k)], i = 1, . . . , n − 1, ¯hn (k + 1) − h¯ n (k) = −τ u¯ n (k) = τ αn (αn−1 + αn )(h¯ n−1 (k) − h¯ n (k)). (16) Let h¯ (k) = [h¯ 0 (k), . . . , h¯ n (k)]T , we have h¯ (k + 1) = P h¯ (k),
(17)
where P = In+1 − τ L˜ with L˜ given by Eq. (18) in Box II. When 2 0 < τ < 1/(4αM ), P is a stochastic matrix with positive diagonal elements and the graph corresponding to P contains a spanning tree. According to Lemma 3, P is SIA. Then, lim h¯ (k) = lim P k−km h¯ (km ) = 1v T h¯ (km ),
k→∞
k→∞
(19)
where v is a column vector with nonnegative elements. Note that h¯ (km ) > 0, i = 0, 1, . . . , n holds based on Lemma 1. Therefore, d (q (k),q (k)) limk→∞ ρ αi +α i+1 = c for i = 0, 1, . . . , n where c is a positive i
i+1
number.
j̸=i
≥ (1 − τ αi Lii )¯ui (k) − τ αi
j̸=i
Lij sat(¯uj (k)),
Now we are ready to present the main result of this paper in the following theorem.
L. Dou et al. / Automatica 81 (2017) 464–470
α (α + α ) 1 2 1 .. . ··· L˜ = .. . 0
−α1 (α1 + α2 ) .. .
··· .. .
0
−αi (αi−1 + αi ) .. .
.. . αi+1 (αi+1 + αi+2 ) + αi (αi−1 + αi ) .. .
0
···
469
−αi+1 (αi+1 + αi+2 ) .. . −αn (αn−1 + αn )
0
.. . ··· .. .
.
(18)
αn (αn−1 + αn )
Box II.
Fig. 1. (a) Time evolution of the homogeneous mobile sensors’ positions qi (i = 1, . . . , 5) on a line; (b) The roughness function ρ(x) = 5ex for x ∈ [0, 1], and the final positions of the homogeneous sensors in [0, 1] on the x-axis.
Fig. 2. (a) Time evolution of the heterogeneous mobile sensors’ positions qi (i = 1, . . . , 5) on a line; (b) The control inputs ui (i = 1, . . . , 5) of the heterogeneous mobile sensors.
2 Theorem 1. Suppose condition (1) is satisfied. If 0 < τ < 1/(4αM ), with the proposed coverage control laws (7), networked mobile sensors will converge to the optimal set of positions which minimizes the coverage cost function T (q1 , . . . , qn , ρ) with order preservation.
Proof. Given the initial condition (1), it has been proved in Lemma 1 that the execution of the proposed control laws (7) always preserves the mobile sensors’ ordering if the sampling 2 period τ satisfies 0 < τ < 1/(4αM ). It follows from Lemma 4 that dρ (qi (k), qi+1 (k))/(αi + αi+1 ) will converge to a same positive constant for all i = 0, 1, . . . , n as time goes into infinity. The necessary and sufficient conditions in Lemma 2 are thus satisfied in the limit, and the mobile sensors converge to the set of positions corresponding to the minimum coverage cost. 5. Simulations In this section, the effectiveness of the proposed coverage control laws (7) is illustrated by simulating a group of 5 mobile sensors. The starting positions qi (0), i = 1, . . . , 5 are randomly chosen in the interval (0, 1), satisfying the initial condition (1). The roughness of the terrain is nonuniform which is described by ρ(x) = 5ex . First, we consider a group of homogeneous mobile sensors with the same maximum actuation given by 0.2. The sampling period is chosen as τ = 0.4. Fig. 1(a) displays the evolution of the sensors’ positions. Fig. 1(b) shows the roughness function ρ(x) for x ∈ [0, 1] and the final positions of the sensors on the x-axis. It can be seen from Fig. 1(b) that in the final configuration more sensors are positioned where the roughness function ρ is higher. Next, we consider a group of heterogeneous mobile sensors with different maximum actuations given by [0.2, 0.2, 0.4, 0.1, 0.8]. We set the sampling period as τ = 0.3. The evolution of the heterogeneous sensors’ positions is shown in Fig. 2(a). It can be
Fig. 3. Time evolution of functions Ti and T for a heterogeneous mobile sensor network.
observed from Figs. 1(a) and 2(a) that the sensors’ spatial ordering is preserved throughout the evolution of the networks. Fig. 2(b) shows the evolution of the sensors’ control inputs. We introduce the variables Ti = dρ (qi , qi+1 )/(αi + αi+1 ), i = 0, . . . , 5 to verify whether the optimality condition in Lemma 2 is satisfied. Fig. 3 shows the time evolution of Ti , i = 0, . . . , 5 and the coverage cost function T (q1 , . . . , q5 , ρ). It can be seen that the variables Ti finally achieve a consensus. Meanwhile, the coverage cost function T (q1 , . . . , q5 , ρ) reaches the minimum. 6. Conclusion This paper has proposed a distributed control scheme for a network of heterogeneous mobile sensors to optimally position themselves on a line with varying roughness. Each sensor is assumed to have a different actuation limit. We have developed distributed coverage control laws with input constraints which guarantee the minimization of the largest time required for the mobile sensor network to reach any point on the line. Moreover, with the proposed control laws, the ordering of the sensors is preserved and
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Liya Dou received the B.S. degree in information and computing science, and M.S. degree in mathematics from University of Science and Technology Beijing, Beijing, China, in 2011 and 2014, respectively. She is now working towards her joint Ph.D. degree with the Department of Mechanical and Biomedical Engineering, City University of Hong Kong, Hong Kong SAR, China, and also with the Department of Automation, Shanghai Jiao Tong University, Shanghai, China. Her research interests include cooperative control and multi-agent networks.
Cheng Song received the B.Eng. degree in Automation and the Ph.D. degree in Control Science and Engineering from the University of Science and Technology of China, Hefei, China, in 2007 and 2012, respectively, and the Ph.D. degree in the Department of Mechanical and Biomedical Engineering from the City University of Hong Kong, Kowloon, Hong Kong, in 2012. He is currently an Associate Professor in the School of Automation, Nanjing University of Science and Technology. His main research interests include cooperative control, multi-agent systems, and mobile sensor networks.
Xiaofan Wang received the Ph.D, degree from Southeast University, China in 1996. He has been a Professor in the Department of Automation, Shanghai Jiao Tong University (SJTU) since 2002 and a Distinguished Professor of SJTU since 2008. He received the 2002 National Science Foundation for Distinguished Young Scholars of P. R. China, the 2005 Guillemin-Cauer Best Transactions Paper Award from the IEEE Circuits and Systems Society, the 2008 Distinguished Professor of the Chang Jiang Scholars Program, Ministry of Education, and the 2015 Second Class Prize of the State Natural Science Award. His current research interests include analysis and control of complex dynamical networks. He is currently the Chair of the IFAC Technical Committee on Large-Scale Complex Systems, Board member of the international Network Science Society (NetSci) and Chair of the Chinese Technical Committee on Complex Networks an System Control.
Lu Liu received her Ph.D. degree in 2008 in the Department of Mechanical and Automation Engineering, Chinese University of Hong Kong, Hong Kong. From 2009 to 2012, she was an Assistant Professor in University of Tokyo, Japan, and then a Lecturer in University of Nottingham, United Kingdom. She is currently an Assistant Professor in the Department of Mechanical and Biomedical Engineering, City University of Hong Kong, Hong Kong. Her research interests are primarily in networked dynamical systems, control theory and applications, and biomedical devices. She received the Best Paper Award (Guan Zhaozhi Award) in the 27th Chinese Control Conference in 2008.
Gang Feng received the B.Eng. and M.Eng. degrees in Automatic Control from Nanjing Aeronautical Institute, China in 1982 and in 1984 respectively, and the Ph.D. degree in Electrical Engineering from the University of Melbourne, Australia in 1992. He has been with City University of Hong Kong since 2000 where he is now Chair Professor of Mechatronic Engineering. He was lecturer/senior lecturer at School of Electrical Engineering, University of New South Wales, Australia, 1992–1999. He was awarded an Alexander von Humboldt Fellowship in 1997, and the IEEE Transactions on Fuzzy Systems Outstanding Paper Award in 2007, and Changjiang chair professorship from Education Ministry of China in 2009. His current research interests include multi-agent systems and control, intelligent systems and control, and networked systems and control. Prof. Feng is an IEEE Fellow, an associate editor of IEEE Trans. Fuzzy Systems and Journal of Systems Science and Complexity, and was an associate editor of IEEE Trans. Automatic Control, IEEE Trans. Systems, Man & Cybernetics, Part C, Mechatronics, and Journal of Control Theory and Applications.