Elsevier

Neurocomputing

Volume 344, 7 June 2019, Pages 28-36
Neurocomputing

Optimal sensor placement based on relaxation sequential algorithm

https://doi.org/10.1016/j.neucom.2018.03.088Get rights and content

Abstract

Aiming at the large tension of the solution obtained by sequential algorithm for optimal sensor placement problem, a novel relaxation sequential algorithm is proposed by introducing the idea of edge relaxation operation of Dijkstra’s algorithm into sequential algorithm. An initial solution set is generated by sequential algorithm, and improved by relaxation till the relaxation operation terminates. The proposed algorithm takes modal assurance criterion (MAC) matrix as the objective fitness function. A truss structure and a rigid-framed arch bridge are applied as examples to verify the effectiveness of the new algorithm for optimal sensor placement. The result indicates that the relaxation sequential algorithm requires fewer sensors and can reach better maximum off-diagonal element of MAC matrix in OSP problem.

Introduction

Large-scale complex structures will be damaged during long service period, especially when they are exposed to harsh environment or natural calamities. Modern technologies for structural safety generally require control systems to monitor the structural behavior during the whole service life cycle. These technologies usually depend on the adoption of increasingly reliable sensors suitable for the monitoring purposes. However, the quality of the obtained information significantly depends on the numbers and positions of corresponding sensors [1]. Owing to the cost limitation, it is difficult and barely possible to place sensors in all appropriate positions. In this sense, deploying fewer sensors on the structures and acquiring more structure health information are key issues. Especially, how to place sensors reasonably becomes one of the most important problems, which is known as optimal sensor placement (OSP).

Due to the above-mentioned reason, the OSP has received considerable attentions and has been investigated in different areas in the past decade. Generally, there are two possible strategies for OSP: the first one is based on sequential algorithm, and the second one is based on evolutionary algorithm.

As far as the evolutionary algorithm-based strategy, the number of sensors is fixed, and only the locations of sensors are needed to be adjusted. For example, Yao et al. has investigated Genetic Algorithm (GA) to place sensors, and the simulation results showed better accuracy than Effective Independence (EFI) method [2]. Guo et al. presented some strategies (crossover based on identification code, mutation based on two gene bits) to improve the convergence speed of simple GA [3]. Liu et al. introduced a GA method, which used the decimal two-dimension array coding method to overcome the low computational efficiency of traditional approaches for determining the optimal placement of sensors [4]. Another improved genetic algorithm known as the generalized genetic algorithm (GGA) is adopted to obtain the optimal placement of sensors. The placement scheme obtained by the GGA showed better feasibility and effectiveness compared with GA [5]. In recent years, the successful applications of new intelligence algorithms in combinatorial optimization problems, such as traveling salesman problem (TSP) and knapsack problem (KP) [6], [7], bring improved ideas to the problem of OSP. Jung et al. presented the application of genetic algorithm to the sensor placement optimization for improving the modal identification quality of flexible structures [8]. He et al. proposed a modified modal assurance criterion (MMAC) to improve the modal energy of the selected locations, and used the improved adaptive genetic algorithm to enhance computation efficiency [9]. Jin et al. presented an improved harmony search (HS) algorithm, and used modal assurance criterion (MAC) to investigate the optimization problem of sensor placement on gantry crane structures [10]. Yi et al. proposed a novel distributed wolf algorithm to improve the optimization performance in identifying the best sensor locations [11].

As for the heuristic algorithms mentioned above, repeated iterations can obtain different solutions which have different qualities. Qualities of those solutions depend on the optimization capacity of algorithms. Sequential algorithm has a more stable optimization capacity compared with different heuristic algorithms, which can only obtain one local optimum but feasible solution. As far as the sequential algorithm-based strategy is concerned, increase (or decrease) iteratively the number of sensors till the termination condition is satisfied. For example, Kammer developed the effective independence method (EFI), which ranks candidate sensor locations based upon their contribution to the linear independence of the target modal partitions [12]. The EFI method tends to remove sensor with the lowest contribution to independence. Finally, a subset of sensor positions is selected as the solution of optimal sensor placement problem. Based on a given rank for the system observability matrix, Lim employed an optimal method capable of determining sensor locations by test constraints [13]. Carne and Dohmann used modal assurance criterion (MAC) matrix as a measure of the utility of a sensor configuration [14]. Heo et al. presented the modal kinetic energy (MKE) method and optimized the transducer placement of a long span bridge for identification and control purposes [15]. Miller introduced an approach to compute a Gaussian quadrature Formula to offer the optimal locations of sensors, and illustrated effectiveness of the proposed method on a slewing beam [16]. Park et al. utilized modal controllability and observability which are defined in balanced coordinate system to select the locations of sensors and actuators [17]. Meo and Zumpano modified the EFI method, and proposed an effective independence driving-point residue (EFI-DPR) method for OSP to identify the vibration characteristics of bridge [18]. Li et al. combined EFI method and MKE method, raised a quick EFI method through QR decomposition, and demonstrated the connections between EFI and MKE on the I-40 Bridge [19].

From the reviews of the methods based on the aforementioned sequential algorithms, it can be concluded that the mathematical models were improved by many researches. However, only few literatures focused on improving the iterative process. Therefore, this paper mainly focuses on improving the iterative process of sequential algorithm for obtaining an improved optimal result.

The purpose of this study is to introduce relaxation technique into traditional sequential algorithm to solve the OSP problem. This new algorithm is termed as relaxation sequential algorithm. That is, relaxation sequential algorithm is inspired by edge relaxation which is the basic operation of Dijkstra’s algorithm [20]. It is generally accepted that the shortest path problem of a graph with edge weights can be effectively solved by Dijkstra’s algorithm.

This paper is organized as follows:

Section 2 gives a description of the relaxation sequential algorithm for OSP problem. Section 3 has two examples that use relaxation sequential algorithm to solve OSP problems. A few concluding remarks are given in Section 4.

Section snippets

Relaxation sequential algorithm

In the OSP problem, a structure is given and the modal matrix ΦRn×m is obtained by implementing modal analysis algorithm; the input of modal analysis algorithm is the finite element model (FEM) of the structure. The goal of OSP is to select k rows from the modal matrix Φ so that the value of objective fitness function can be as optimal as possible. The maximum off-diagonal element of MAC matrix is selected as the objective fitness function, which has been frequently used to measure the

Case 1

To verify the effectiveness of the proposed relaxation sequential algorithm described in Section 2, we apply the proposed method and the forward sequential algorithm to the OSP problem for a steel truss model, which is shown in Fig. 2. The weight of steel truss model is 54 kilograms. The size of the structure is 2800 × 360 × 270 (L ×  W ×  H) millimeters. The structure is composed of seven sections; each section is 400 mm long. Elastic modulus of steel truss is E=1.9 × 1011  Pa, and Poisson’s

Computational complexity analysis

Complexity is also an important factor in analysis of algorithms. Therefore, complexity of relaxation sequential algorithm is analyzed to explore the complexity level of algorithm.

If an algorithm is composed of several parts, then its complexity is the sum of the complexities of these parts. The algorithm may consist of a loop executed many times, and each time is with a different complexity. We say that the complexity of the algorithm, or the running time, namely O(f(N)) [22].

The computational

Conclusion

Considering the large tension of the solution obtained by sequential algorithm, relaxation sequential algorithm is adapted to solve the OSP problem in this paper. The sequential algorithm is modified by referring to the idea of edge relaxation. Steel truss model and rigid-framed arch bridge model are employed as the objectives to verify the proposed approach. The first 6-order modal shapes of steel truss and first 10-order modal shapes of rigid-framed arch bridge are extracted using ANSYS

Acknowledgment

This research is supported by National Natural Science Foundation of China (nos. 61463028, 51768035).

Hong Yin, she was born in Xinjiang Uygur Autonomous Region, China, in 1978. She received the B.S. degree from Lanzhou Jiaotong University, Lanzhou, in 2000, in mechanical engineering. She received the M.S. degree from Lanzhou Jiaotong University, Lanzhou, in 2003, in vehicle engineering. She is currently pursuing the Ph.D. degree with the school of mechatronicengineering, Lanzhou Jiaotong University. Her research interests include measurement and control, intelligence optimization.

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    Hong Yin, she was born in Xinjiang Uygur Autonomous Region, China, in 1978. She received the B.S. degree from Lanzhou Jiaotong University, Lanzhou, in 2000, in mechanical engineering. She received the M.S. degree from Lanzhou Jiaotong University, Lanzhou, in 2003, in vehicle engineering. She is currently pursuing the Ph.D. degree with the school of mechatronicengineering, Lanzhou Jiaotong University. Her research interests include measurement and control, intelligence optimization.

    Kangli Dong, he was born in Shandong Province, China, in 1994. He received the B.S. degree from Shandong University of Technology, Zibo, in 2016, in mechanical design manufacture and automation. He is currently pursuing the M.S. degree with the school of mechatronic engineering, Lanzhou Jiaotong University. His research interests include intelligent optimization, optimal sensor placement.

    An Pan, he was born in Gansu Province, China, in 1986. He received the B.S. degree from North University of China, Taiyuan, in 2008, in optical information science and technology. He research interests include optimal sensor placement.

    Zhenrui Peng, he was born in Gansu Province, China, in 1972. He received the B.S. degree and M.S. degree from Lanzhou Jiaotong University, Lanzhou, in 1995 and in 2001 respectively, both in mechanical engineering. He received the Ph.D. degree from Zhejiang University, Hangzhou, in 2007, in control science and engineering. He is now with the school of mechatronic engineering, Lanzhou Jiaotong University. His research interests include measurement and control, intelligence optimization.

    Zhaoyuan Jiang, he was born in 1954. He received the B.S. degree from Lanzhou Jiaotong University, Lanzhou, in 1982. He received the M.S. degree and Ph.D. degree from Southwest Jiaotong University, Chengdu, in 1988 and in 1997, respectively.

    Shaoyuan Li, he was born in Hebei, China, in 1965. He received the B.S. and M.S. degrees in automation from Hebei University of Technology, Tianjin, China, in 1987 and 1992, respectively, and the Ph.D. degree from the Department of Computer and System Science, Nankai University, Tianjin, in 1997. Since July 1997, he has been with the Department of Automation, Shanghai Jiao Tong University, Shanghai, China, where he is currently a Professor. His research interests include fuzzy systems, model predictive control, dynamic system optimization, and system identification.

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