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Radial Basis Function Neural Network in Vibration Control of Civil Engineering Structure

Data publikacji: 15 Jul 2022
Tom & Zeszyt: AHEAD OF PRINT
Zakres stron: -
Otrzymano: 28 Jan 2022
Przyjęty: 17 Mar 2022
Informacje o czasopiśmie
License
Format
Czasopismo
eISSN
2444-8656
Pierwsze wydanie
01 Jan 2016
Częstotliwość wydawania
2 razy w roku
Języki
Angielski
Abstract

This article uses the radial basis function artificial neural network and the MATLAB toolbox to study the vibration control of civil engineering structures. The article proposes a dynamic structure design method based on a generalized radial basis function neural network. Furthermore, the RBF neural network theory is used to optimize the structure-related controller parameters in geotechnical engineering. The research results show that RBF neural network can more accurately predict the vibration response of civil engineering. It can effectively solve the time lag problem in vibration control.

Keywords

MSC 2010

Introduction

The use of active/semi-active intelligent control systems for the building structure can significantly suppress its vibration response. This can meet higher safety and functional requirements. The main idea is to adopt the traditional centralized control scheme of “distributed collection and centralized processing.” However, suppose the complex engineering structure with large scale, and multiple functions still adopt traditional centralized control. In that case, it will inevitably face the following problems: 1) The information exchange of the control system is extremely complicated and easily causes lag. 2) Control system integration and operating costs increase. 3) Once an individual sensor, actuator, or control platform fails, the system's reliability will decrease or even fail. At the same time, the uncertainty of external loads such as earthquakes and typhoons and the structure itself has increased the fluctuation of the control effect of traditional centralized control schemes to a considerable extent.

Because of the above reasons, many scholars at home and abroad have introduced large-scale system decentralized control theory into the field of civil engineering and have done corresponding theoretical research. It is mainly aimed at the application of different decentralized control algorithms in structural vibration control. Some scholars have studied the robust decentralized control method considering time delay and verified the method. However, the distributed control system's distributed controllers work in parallel, and there is no subordination relationship, so it isn’t easy to carry out effective coordination [1]. At the same time, each controller has its control target. If conflicts occur, they need to be resolved by game theory. Therefore, studying the hierarchical decentralized control of building structure is necessary because of the above problems. In this multi-level control system, the overall problem is solved by the lower-level local controller. The higher level performs coordinated control based on considering associations and constraints to achieve global optimization. Some scholars have studied the hierarchical decentralized robust control design method of uncertain building structures and verified the effectiveness through the cantilever beam vibration test.

This paper studies the hierarchical decentralized control of building structures. Based on the coordinated control of the global controller, an adaptive RBF neural network control law for local control is derived. We use a differential evolution algorithm to optimize the local sub-controller [2]. The article finally established an adaptive RBF neural network hierarchical decentralized control algorithm suitable for structural vibration control.

Hierarchical decentralized control theory
State equation of traditional centralized control system

The motion equation of the n story building under seismic excitation is: MX+C¯X+KX=BsU+EsxgX(t0)=X0X(t0)=X0 \matrix{{MX + \bar CX + KX = {B_s}U + {E_s}{x_g}} \hfill \cr {\matrix{{X({t_0}) = {X_0}} & {X({t_0}) = {X_0}} \cr}} \hfill \cr} X is the displacement of the structure relative to the ground. M, C¯ \bar C and K are the mass, damping and stiffness matrices of the structure, respectively. U is control. Bs is the position matrix of the actuator. Es is the external excitation position matrix.

Define the state variable z = {X X}T. Then the original equation of motion (1) can be expressed as the following state equation form: z=Az+Bu+Exgy=Cz+Du+Fx¯g \matrix{{z = Az + {B_u} + E{x_g}} \hfill {y = Cz + {D_u} + F{{\bar x}_g}} \hfill } Where: A=[0IM1KM1C¯]2n×2nB=[0M1Bs]2n×pE={0n×11n×1} \matrix{{A = {{\left[{\matrix{{\matrix{0 \hfill & I \hfill }} \hfill {- {M^{- 1}}K - {M^{- 1}}\bar C} \hfill }} \right]}_{2n \times 2n}}\,\,B = {{\left[{\matrix{0 \hfill {- {M^{- 1}}{B_s}} \hfill }} \right]}_{2n \times p}}} \hfill {E = \left\{{\matrix{{{0_{n \times 1}}} \hfill {- {1_{n \times 1}}} \hfill }} \right\}} \hfill } n is the structural dimension. p is the number of actuators in the structure. y is the observation output, and its dimension is determined according to the observation requirements. C, D and F are the observation output matrices of appropriate dimensions.

State equation of hierarchical decentralized control system

Suppose there are two local sub-controllers and one global controller after the decomposition of the original control system (as shown in Figure 1). Each sub-controller completes the vibration control of the corresponding subsystem only according to the feedback information of the subsystem [3]. The global controller is used to eliminate the coupling between subsystems. The state equation of any subsystem can be expressed as: zi=Aizi+Biui+Eixg+jiNAijzj {z_i} = {A_i}{z_i} + {B_i}{u_i} + {E_i}{x_g} + \sum\limits_{j \ne i}^N {{A_{ij}}{z_j}} i = 1 … N.Ai, Bi and Ei are the system matrix, control force matrix and external excitation matrix of the corresponding subsystem respectively. Aij. The associated coupling item of Subsystem i and Subsystem j. zi=[xi;x¯i]2ni×1 {z_i} = {[{x_i};{\bar x_i}]_{2{n_i} \times 1}} is the displacement and velocity response of subsystem i. Ai=[0IMi1KiMi1Ci]2ni×2ni {A_i} = {\left[{\matrix{0 & I {- M_i^{- 1}{K_i}} & {- M_i^{- 1}{C_i}} }} \right]_{2{n_i} \times 2{n_i}}} , Bi=[0Mi1Bsi]2ni×pi {B_i} = {\left[{\matrix{0 \hfill {- M_i^{- 1}{B_{{s_i}}}} \hfill }} \right]_{2{n_i} \times {p_i}}} , Ei={0ni×11ni×1} {E_i} = \left\{{\matrix{{{0_{{n_i} \times 1}}} \hfill {- {1_{{n_i} \times 1}}} \hfill }} \right\} , Mi, Ki and Ci are the mass, stiffness and damping matrices of the subsystem i. Bsi is the actuator position matrix of the subsystem i. ni and pi are the dimension of subsystem i and the number of actuators.

Figure 1

Hierarchical distributed control system

According to the hierarchical decentralized control theory, the control force can be expressed as: ui=uil+uig {u_i} = u_i^l + u_i^g uil u_i^l represents the local control power, and uig u_i^g is the global control power. uil=Gilzl u_i^l = - G_i^l{z_l} , uig=j=1,jiN u_i^g = - \sum\limits_{j = 1,j \ne i}^N {} , HjGigzj {H_j}G_i^g{z_j} . Among them Gil G_i^l and Gig G_i^g are local sub-controllers and global controllers, respectively. Hj is the conversion matrix [4]. To obtain the global control force related only to the adjacent layers between the subsystems, we substitute it into equation (3) as follows: zi=(AiBiGil)zi+Eixg+j1N(AijBiHjGig)zj {z_i} = ({A_i} - {B_i}G_i^l){z_i} + {E_i}{x_g} + \sum\limits_{j \ne 1}^N {({A_{ij}} - {B_i}{H_j}G_i^g){z_j}} The global controller can be obtained by using AijBiHjGig=0 {A_{ij}} - {B_i}{H_j}G_i^g = 0 : Gig=((BiHj)TBiHj)1(BiHj)TAij G_i^g = {({({B_i}{H_j})^T}{B_i}{H_j})^{- 1}}{({B_i}{H_j})^T}{A_{ij}} The local sub-controller is determined by the RBF network adaptive control law derived below.

Design of an adaptive RBF neural network local sub-controller

The Radial Basis Function (RBF) network simulates the neural network structure of the human brain that is locally adjusted and covers the receptive field. It has been proved that the RBF network can approximate any continuous function with arbitrary precision. The RBF network is a 3-layer feed-forward network with a single hidden layer [5]. The basis function output from the input layer to the hidden layer is a nonlinear mapping. The output is linear and locally approximating. Therefore, it meets the requirements of real-time control. The RBF network structure of multiple input single output is shown as in Fig. 2.

Figure 2

RBF neural network structure

In the RBF neural network in this paper, zi = [xi, xi]2ni×1 is the network input. h = [h1, …, hm]T, hj is the output of the j neuron, namely hj=exp(zicj22bj2) {h_j} = \exp \left({- {{{{\left\| {{z_i} - {c_j}} \right\|}^2}} \over {2b_j^2}}} \right) j = 1,2, …, m, cj = [cj1, …, cjm] is the vector value of the center point of the j hidden layer neuron. bj > 0 is the width of the Gaussian function of the neuron j. The network weight is w = [ω1, L, ωm]T. The output of the RBF network is: f^=wTh \hat f = {w^T}h

Local sub-controller optimization based on differential evolution algorithm

From the expression of the Gaussian function, it can be seen that the Gaussian function is affected by the parameters cj and bj. Among them, the width of the Gaussian function is an important factor that affects the range of network mapping. The larger the value of bj, the greater the ability of the network to map the input [6]. The closer the center point vector cj of the Gaussian function is to the input, the more sensitive the Gaussian function is to the input. Appropriate Gaussian function parameters and adaptive law coefficients must be selected to improve the mapping ability of the adaptive law of the RBF neural network. Differential evolution algorithm is an optimization algorithm based on swarm intelligence theory. It reorganizes to obtain the intermediate population through the differences of the individuals in the current population. It then uses the fitness value of the direct father-son mixture of individuals to obtain the new generation population. Standard differential evolution algorithm unique variation formula: εik+1=εr1k+Γ(εr2kεr3k) \varepsilon _i^{k + 1} = \varepsilon _{r1}^k + \Gamma (\varepsilon _{r2}^k - \varepsilon _{r3}^k) ε is an individual in the population, and the subscript ri is a random integer. It represents the serial number of the individual in the population. ( εr2kεr3k \varepsilon _{r2}^k - \varepsilon _{r3}^k ) is the differentiation vector. Γ is the variation factor. The purpose of drawing on the best individual information in the current population ( pgk p_g^k ) is to speed up the convergence speed of the differential evolution algorithm. We modify equation (9) as εik+1=pgk+Γ(εr2kεr3k) \varepsilon _i^{k + 1} = p_g^k + \Gamma (\varepsilon _{r2}^k - \varepsilon _{r3}^k) The optimization process of the differential evolution algorithm is shown in Figure 3. The constraint condition of the optimization objective is maxj|Fj(t)|800kN \mathop {\max}\limits_j |{F_j}(t)| \le 800kN (a single actuator output limit). In this paper, a penalty function is used to deal with the above constraints to optimize the objective function defined as follows Jopt=max{|xic|}max{|xiuc|}+Pfmax[0,max{|Fj|}Fmax1] {J_{opt}} = {{\max \{|x_i^c|\}} \over {\max \{|x_i^{uc}|\}}} + Pf\max [0,{{\max \{|{F^j}|\}} \over {{F_{\max}}}} - 1] xic x_i^c and xiuc x_i^{uc} respectively represent the relative displacement of the i level structure of the controlled structure and the uncontrolled structure. The second term takes into account the constraints, and Pf represents the penalty factor. Just take a larger value, this item is zero when the constraint condition is satisfied, and the objective function value is larger when the constraint condition is not satisfied. The probability of its being selected is very small.

Figure 3

DE optimization flowchart

Simulation Analysis
Benchmark model

We select the 9-story steel structure Benchmark model designed by ASCE as a simulation example. We use the static cohesion method to reduce the order of the original finite element model and only retain nine translational degrees of freedom. To verify the proposed adaptive RBF neural network hierarchical decentralized control algorithm, two far-field seismic waves and one near-field seismic wave are selected: ElCentro wave, Hachinohe wave, and Kobe wave seismic excitation [7]. The duration is the 40s, and the peak value is 300cm/s2. Actuators are arranged on each layer of the structure, and the maximum output of a single actuator is 800kN. In the case of this paper, the centralized control condition adopts the LQG control algorithm and uses the differential evolution algorithm to optimize the controller. We take the average of the optimization results under the above seismic excitation as the optimal controller parameters to ensure the optimal centralized control effect. The simplified model of the original structure is shown in Figure 4.

Figure 4

Simplified structure model diagram and schematic diagram of structure hierarchical decentralized control

Local sub-controller optimization

The hierarchical distributed control system designed in this paper includes a global controller and three local sub-controllers and k1 = diag([100 100 100]), k2 = diag([200 200 200]), Q = diag([10 10 10 10 10 10]) is selected for any local sub-controller. We choose an RBF network with a structure of 6-5-1. To simplify the optimization process of the local sub-controller, we choose the following parameters to be optimized: cjx = λ1[−0.1 −0.050 0.05 0.1], cjẋ = λ1[−2.5 −1.30 1.32 2.5], bj = λ3. Also, note the adaptive law coefficient γ = λ4. The parameter selection of the differential evolution algorithm is shown in Table 1. Figure 5 shows the convergence curve of the objective function for hierarchical decentralized control optimization (taking the average of three seismic waves). Figure 6 shows the optimization process of RBF network parameters and adaptive law influence coefficients. For the convenience of drawing, we normalize λ4. The optimal parameter is λ1 = 0.5, λ2 = 0.077, λ3 = 0.109, λ4 = 100.

Figure 5

Convergence curve of the objective function

Figure 6

Optimization process of controller parameters

DE algorithm parameters

The maximum number of iterations 30
Population dimension 50
Variation factor 0.3
Crossover factor 0.6
Numerical analysis of hierarchical decentralized control

The time history response curves of the displacement between the bottom layers of the structure under the ElCentro seismic wave and the hierarchical decentralized control and the control force-time history curves of the bottom actuators are shown in Figure 7. The peak response of the first-story displacement of the hierarchical decentralized control structure is significantly smaller than the response under centralized control [8]. At the same time, the control effect under the hierarchical decentralized control within the duration of the ground motion is better than the centralized control. Without significantly increasing the control force, the adaptive RBF network hierarchical decentralized control algorithm can make the actuator function more efficiently. Therefore, it can better approximate the desired response of the structure.

Figure 7

Time history of displacement between bottom layers and time history of top actuator control force

The maximum interstory displacement angle and absolute acceleration response of the uncontrolled structure, centralized control, and hierarchical decentralized control under ElCentro seismic excitation are shown in Fig. 8. The hierarchical decentralized control based on the adaptive RBF neural network algorithm can well suppress each floor of the structure [9]. The displacement angle of each floor of the structure has a better control effect than the centralized control. At the same time, the absolute acceleration control effect of each floor of the structure is similar to centralized control. The control effect of some floors has been improved.

Figure 8

The maximum displacement angle and maximum absolute acceleration of the structure

Table 2 shows the evaluation index values obtained by using the adaptive RBF neural network hierarchical decentralized control algorithm under different seismic excitations and the average value of the evaluation index obtained by the centralized control LQG algorithm. Among them J1 ~ J3 is the peak value of the displacement angle between the layers of the structure, the peak value of the absolute acceleration, and the peak value of the base shear force. J4 ~ J6 is the peak value of the displacement angle between the layers, the absolute acceleration, and the base shear force norm. J7 is the self-defined evaluation index in this paper to reflect structural damping energy consumption: J7=iEDiiEDiuc {J_7} = {{\sum\limits_i {E{D_i}}} \over {\sum\limits_i {E{D_{iuc}}}}} EDi and EDiuc represent the damping energy of layer B of uncontrolled structure and controlled structure, respectively.

Related evaluation indicators of structural response

ARBFHDCS LQG average
E K H average value
J1 0.68 0.69 0.71 0.69 0.74
J2 0.68 0.73 0.75 0.72 0.74
J3 0.46 0.32 0.52 0.43 0.46
J 0.56 0.58 0.77 0.64 0.75
J5 0.52 0.48 0.7 0.57 0.65
J6 0.58 0.6 0.78 0.65 0.76
J7 0.31 0.33 0.53 0.39 0.52

It can be seen from Table 2 that the evaluation indexes of hierarchical decentralized control under Kobe and Hachinohe excitation all show similar results to the conclusions described above. Compared with centralized control, the average value of each hierarchical decentralized control evaluation index has been improved to different degrees. Among them, the control effect of J4 ~ J7 has improved significantly [10]. It increased respectively: 14.7%, 12.6%, 13.7% and 24.7%. After the global controller eliminates the correlation and coupling between the subsystems, each local sub-controller only depends on the corresponding subsystem's displacement and velocity response feedback information. There is no need for any information exchange between the sub-controllers, which shortens the transmission time of information feedback and control instructions. This reduces the time lag and forms an efficient, independent, and autonomous local sub-controller. This ensures that the actuators in the subsystems function more fully. Therefore, the hierarchical decentralized control strategy designed by the adaptive RBF neural network algorithm shows a better shock absorption effect.

Conclusion

Based on the advantages of adaptive control, RBF neural network, and hierarchical decentralized control, this paper proposes a hierarchical decentralized control algorithm suitable for structural vibration control. At the same time, we use a differential evolution algorithm to optimize the RBF neural network parameters and adaptive law influence coefficients. The algorithm can effectively control the seismic response of the structure, which shows that the ARBFHDC algorithm in this paper is effective and feasible. The adaptive RBF neural network hierarchical decentralized control optimized by the differential evolution algorithm can obtain a better shock absorption effect than the centralized control.

Figure 1

Hierarchical distributed control system
Hierarchical distributed control system

Figure 2

RBF neural network structure
RBF neural network structure

Figure 3

DE optimization flowchart
DE optimization flowchart

Figure 4

Simplified structure model diagram and schematic diagram of structure hierarchical decentralized control
Simplified structure model diagram and schematic diagram of structure hierarchical decentralized control

Figure 5

Convergence curve of the objective function
Convergence curve of the objective function

Figure 6

Optimization process of controller parameters
Optimization process of controller parameters

Figure 7

Time history of displacement between bottom layers and time history of top actuator control force
Time history of displacement between bottom layers and time history of top actuator control force

Figure 8

The maximum displacement angle and maximum absolute acceleration of the structure
The maximum displacement angle and maximum absolute acceleration of the structure

DE algorithm parameters

The maximum number of iterations 30
Population dimension 50
Variation factor 0.3
Crossover factor 0.6

Related evaluation indicators of structural response

ARBFHDCS LQG average
E K H average value
J1 0.68 0.69 0.71 0.69 0.74
J2 0.68 0.73 0.75 0.72 0.74
J3 0.46 0.32 0.52 0.43 0.46
J 0.56 0.58 0.77 0.64 0.75
J5 0.52 0.48 0.7 0.57 0.65
J6 0.58 0.6 0.78 0.65 0.76
J7 0.31 0.33 0.53 0.39 0.52

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