Mathematical Problems in Engineering

Volume 2017, Article ID 7213125, 11 pages

https://doi.org/10.1155/2017/7213125

## Optimal Piezoelectric Actuators and Sensors Configuration for Vibration Suppression of Aircraft Framework Using Particle Swarm Algorithm

^{1}School of Electrical Information Engineering, Henan Institute of Engineering, Henan 451191, China^{2}School of Mechatronics Engineering and Automation, Shanghai University, Shanghai 200072, China^{3}School of Computer, Henan Institute of Engineering, Henan 451191, China

Correspondence should be addressed to Huayan Pu; nc.ude.uhs@1002_doogyhp

Received 6 February 2017; Revised 16 April 2017; Accepted 27 April 2017; Published 15 June 2017

Academic Editor: Xiao-Qiao He

Copyright © 2017 Quanzhen Huang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Numbers and locations of sensors and actuators play an important role in cost and control performance for active vibration control system of piezoelectric smart structure. This may lead to a diverse control system if sensors and actuators were not configured properly. An optimal location method of piezoelectric actuators and sensors is proposed in this paper based on particle swarm algorithm (PSA). Due to the complexity of the frame structure, it can be taken as a combination of many piezoelectric intelligent beams and L-type structures. Firstly, an optimal criterion of sensors and actuators is proposed with an optimal objective function. Secondly, each order natural frequency and modal strain are calculated and substituted into the optimal objective function. Preliminary optimal allocation is done using the particle swarm algorithm, based on the similar optimization method and the combination of the vibration stress and strain distribution at the lower modal frequency. Finally, the optimal location is given. An experimental platform was established and the experimental results indirectly verified the feasibility and effectiveness of the proposed method.

#### 1. Introduction

Numbers and locations of piezoelectric sensors and actuators are of great significance for piezoelectric smart structure, vibration control performance, system implementation costs, and so on [1, 2].

There are a few problems for configurations of sensors or actuators in the progress of promoting the engineering application. When they are placed at improper locations, this will lead to much more uncontrolled modal response messages in sensor signals and actuators may inspire uncontrolled modal responses, which will make observation or control spillover or even make the control system unstable. In fact, it is an optimization problem to determine the best locations of sensors and actuators. And many scholars have made some profound studies on this issue. These studies can be split into two: on the one hand, those that determine the optimal allocation criterion, namely, optimizing objective function. On the other hand, those that choose the appropriate optimization methods. For example, Caruso et al. [3] aimed at maximizing the modal controllability and observability of the structure and the locations of piezoelectric patches with a fixed size were optimized. Kumar and Narayanan [4, 5] worked out the optimal location of sensor-actuator pairs which were placed in the piezoelectric cantilever plate on the basis of a linear quadratic regulator (LQR). The LQR performance index was used as the objective function of the optimization progress, which was solved by genetic algorithm (GA). Rao et al. [6] proposed particle swarm based evolutionary optimization technique for optimal placement of piezoelectric patch actuators and accelerometer sensors to suppress vibration. Viswamurthy and Ganguli [7, 8] proposed a response surface based optimization method for actuator and sensor placement. Manning [9] proposed a two-stage optimization strategy for active member placement and strut cross section and compensator parameter optimization for intelligent trusses. T.-W. Kim and J.-H. Kim [10] optimized the placement of the piezoelectric path of the flexible plates by applying the sequential quadratic programming method. Gao [11] investigated integrated optimization of the actuator location and feedback gain in PZT smart trusses with stochastic structural parameters using a two-step optimization strategy. Xian [12] combined a layered strategy and an approximation concept and formed a two-level, branched and multipoint approximation strategy for adaptive truss actuator placement optimization. Liu and Lin [13] took a two-level optimization method based on a simulated annealing algorithm to determine the optimal channel distribution and the optimal channel voltage for dynamic shape control of structures using piezoelectric materials. Honda et al. [14] employed the placement of piezoelectric actuators, the lay-up configurations of laminated composite plates, and the H2 vibration control system as design variables and optimized them simultaneously by GA. Dutta et al. [15] considered artificial bee colony and glowworm swarm optimization algorithms, to find the optimal locations of actuators/sensors and feedback gains of a cantilevered beam.

A piezoelectric aluminum alloy smart frame structure is taken as an experimental model in this paper. Since the structural shape and constraints are complex, the entire framework is taken as a combination of a plurality of piezoelectric intelligent beams and L-shaped structures. Firstly, piezoelectric cantilever is taken as a structural vibration control object. Optimization objective function is established according to controllability and observability criteria of system. Configuration of sensors and actuators is optimized by particle swarm algorithm. Secondly, according to the above optimization process and the analysis results by ANSYS, the final optimal configuration is obtained. An experimental platform was established and the experimental results indirectly verified the feasibility and effectiveness of the proposed method.

#### 2. Optimization Principle of the Piezoelectric Sensors and Actuators

According to Ahari [16], the Partial Differential Equation (PDE) of the cantilever beam which is made of aluminum alloy can be described as follows:

Here is the distribution of external force; is deflection of beam, which is a function of a space variable and time ; is the hardness distribution of the system, which is a partial differential function about a relative space coordinate is a mass density function, which is a positive definite function about the location of . Here, .

The principle for the distribution of the actuators should ensure its influence on the structural perturbation to the greatest degree. In other words, the actuator should transport the energy as large as possible to the structural mode. According to this principle, the criteria for the optimization of the actuators are [17]

is mathematical expectation of the* i*th modal total energy. It can be considered as two parts: the first part is about the total energy of the system. Generally speaking, some low modal energy is chosen on the basis that modal energy reduces sharply along with the increasing of modal numbers; the other part is considered as the volume of an ellipsoid. The ellipsoid is* n*-dimensional, and its radius is directly proportional to each modal energy.

As the size of the diagonal elements of observability Gramian is directly proportional to the stability of the system, the criteria of the optimal sensor placement are

is the eigenvalue of observability Gramian. The eigenvalue is fundamentally the same as that of controllability Gramian for some small damping structural system, so it is the best situation for the same numbers and positions of sensor and actuator.

#### 3. Sensor/Actuator Optimization Algorithm Based on Particle Swarm

##### 3.1. Establishment of Optimization Objective Function

Based on the position optimization guidelines in Section 2, it can be seen that the controllability Gramian gives the relevance of system state and the output and the observability Gramian gives the relevance of the system state and input. But these matrices and eigenvalues are taken depending on the state vector; if the state vector changes, the above will not be established. The above guidelines also show that the vectors of the controllability and observability Gramian are related to optimization configuration criteria; and the controllability and observability Gramian are, respectively, related to matrices* B* and* C*, so matrices* B* and* C* are, respectively, related to the positions of the actuator and sensor.

Therefore, the optimization criterion is to seek the best position to make features of the controllability and observability Gramian optimal. To this end, establish the following best guidelines: when its mathematical value is small, the system is not controllable or not considerable; when the mathematical value is large, the system is controlled or considerable. The value is maximum, and it is the best location.

Before setting the optimal objective function, the observability matrix and the controllability matrix are, respectively, obtained by the expression between Gramian matrix and the system modal energy [18].

Here, is the extraneous signal, is the control signal, and is the deflection of beam.

If the natural frequency distribution of structural vibration system is well and the damping coefficient is small, the optimal objective function is given as follows: where is Gramian matrix, its value is or , is standard deviation for the eigenvalues of Gramian matrix , is the geometric mean of the eigenvalues, its physical significance is the volume of the ellipse, is the coefficient of freedom degree, is the location mainly avoiding both great and small eigenvalues, and is the output energy of the actuator.

can be expressed as each order modal energy or modal value; maximum and minimum values of , , and have synchronization. And may represent sum of each modal energy; a small value can be ignored during the total sum, so that it can be expressed as sum of lower modal energies. can be expressed as product of the modal energies, as we all know that each plays a role in the product operation; it means that all modals are functioning, also taking into account higher-order modals. Therefore, this criterion well considers each order modal; it will be very effective to optimize the configuration for the sensor/actuator.

##### 3.2. Process and Analysis of PSO

According to the characteristics analysis of piezoelectric aluminum alloy beam structure, optimization objective function is obtained for the piezoelectric sensor and actuator. As in literature [2], on the basis of finite element modeling, it is studied on the actuator position optimization of dimensional flexible plate using controlled Gramian matrix. In the process of position optimization using controllability and observability Gramian, each iteration’s calculations need to solve Lyapunov equations; its computation will see a sharp rise when the system has a large degree of freedom or many actuators need to be position-optimized, resulting in a very long computing time. Thus, in order to avoid complex calculations, particle swarm algorithm was proposed in this paper to optimize the objective function, to achieve advantages of high computational efficiency, fast convergence, and being simple and versatile.

Optimization process of piezoelectric sensor/actuator by particle swarm algorithm broadly is divided into several steps:(1)First, establish kinetic equations according to optimization object; expressions of each order’s natural frequencies and modal strain are obtained.(2)Carry out modal strain analysis, and get each order’s natural frequencies, modal shapes, and modal stress distribution.(3)The given modal values and the natural frequency are converted to the needed form of the objective function optimization and then are substituted into the objective function, the position based on particle swarm algorithm is optimized, and ultimately an optimal layout plan of piezoelectric sensor/actuator is given.

###### 3.2.1. Modal Analysis of Smart Piezoelectric Beams

Since it is more complex to analyze directly piezoelectric frame structure modal, in order to more clearly describe the optimization methods in the paper, the constituent units of the piezoelectric frame structure (i.e., piezoelectric beam) are directly analyzed. The material of the framework is aluminum alloy, and its related parameters are as follows: length mm, width mm, height mm, density kg/m^{3}, Young’s modulus Pa, and Poisson’s ratio . Material type of piezoelectric sensors is P-51, material type of piezoelectric actuator is PZT-5H, and its parameters are shown in Table 1 [19].