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Multi-objective optimal design of fuzzy controller for structural vibration control using Hedge-algebras approach

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Abstract

In this paper, the problem of multi-objective optimal design of hedge-algebras-based fuzzy controller (HAC) for structural vibration control with actuator saturation is presented. The main advantages of HAC are: (i) inherent order relationships among linguistic values of each linguistic variable are always ensured; (ii) instead of using any fuzzy sets, linguistic values of linguistic variables are determined by an isomorphism mapping called semantically quantifying mapping (SQM) based on a few fuzziness parameters of each linguistic variable and hence, the process of fuzzy inference is very simple due to SQM values occurring in the fuzzy rule base and (iii) when optimizing HAC, only a few design variables which are above fuzziness parameters are needed. As a case study, a HAC and optimal HACs (opHACs) based on multi-objective optimization view point have been designed to active control of a benchmark structure with active bracing system subjected to earthquake excitation. Control performance of controllers is also discussed in order to shown advantages of the proposed method.

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Abbreviations

ABS:

Active bracing system

CPU time:

Computation time

FC:

Conventional fuzzy controller

GA:

Genetic algorithm

HA:

Hedge algebras

HAC:

Hedge-algebras-based fuzzy controller

L :

Little

LQR:

Linear quadratic regulator

MBBC:

Modified bang–bang controller

Ne :

Negative

opHAC:

Optimal hedge-algebras-based fuzzy controller

\({ op}\hbox {HAC}i\_\mathrm{El}\) :

Optimal hedge-algebras-based fuzzy controllers using the training data from the El Centro earthquake

\({ op}\hbox {HAC}i\_\mathrm{Im}\) :

Optimal hedge-algebras-based fuzzy controllers using the training data from the Imperial Valley earthquake

\({ op}\hbox {HAC}i\_\mathrm{No}\) :

Optimal hedge-algebras-based fuzzy controllers using the training data from the Northridge earthquake

Po :

Positive

PSO:

Particle swarm optimization

SQM:

Semantically quantifying mapping

SQS:

Semantically quantifying surface

SSMC:

Saturated sliding mode controller

V :

Very

AX :

HA structure of a linguistic variable with its term-set X

[a, b]:

Reference domain of a linguistic variable

\(a_{0}, b_{0}\) :

Boundary values of reference domains of the state variables

C :

Set of specific constants

[C]:

Structural damping matrix

\(c^{-}\) :

Set of negative primary terms

\(c^{+}\) :

Set of positive primary terms

\(c_{i}\) :

Damping of the \(i{\mathrm{th}}\) floor

\(c_{0}\) :

Boundary value of reference domain of the control variable

\(d_{i}(t)\) :

Storey drift of the \(i{\mathrm{th}}\) floor in the controlled response

\(d_{\max }\) :

Peak storey drift in the uncontrolled response

\(\{\delta \}\) :

Coefficient vector for earthquake ground acceleration

\(\Delta t\) :

Time step size

\(F_{1}\) :

Peak storey drift

\(F_{2}\) :

Peak absolute acceleration

\(F_{3}\) :

Average control force

\({ fm}(c^{-})\) :

Fuzziness measure of \(c^{-}\)

G :

Set of primary terms

\(g_{X}\) :

Normalization mapping

\(g_{X}^{-1}\) :

De-normalization mapping

H :

Set of hedges

\(H^{-}\) :

Set of negative hedges

\(H^{+}\) :

Set of positive hedges

\(h^{-}\) :

Negative hedge

[K]:

Structural stiffness matrix

\(k_{i}\) :

Atiffness of the \(i{\mathrm{th}}\) floor

[M]:

Structural mass matrix

m :

Number of cycles of the whole control process

\(m_{i}\) :

Mass of the \(i{\mathrm{th}}\) floor

\(M_{up},M_{low}\) :

Boundary values of reference domains of the variables in SQM domain

\(\mu (h^{-})\) :

Fuzziness measure of \(h^{-}\)

\(\varphi \) :

SQM values

T :

Total time of simulation

\(\{U\}\) :

Vector of control forces

u :

Control force

\(u_{\max }\) :

Limitation of the actuator

X :

Term-set of a linguistic variable

\(\{x\}\) :

Vector of the degrees of freedom of the structure

\(\ddot{x}_0 \) :

Ground acceleration

\(\ddot{x}_{ai} (t)\) :

Absolute acceleration of the \(i{\mathrm{th}}\) floor in the controlled response

\(\ddot{x}_{a\max }\) :

Peak absolute acceleration in the uncontrolled response

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Acknowledgements

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Number “107.01-2015.10”.

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Authors and Affiliations

  1. Faculty of Mechanical Engineering, Electric Power University, No. 235 Hoang Quoc Viet Street, Hanoi, Vietnam

    Van-Binh Bui

  2. Faculty of Electro-Mechatronics, Phuong Dong University, No. 171, Trung Kinh Street, Hanoi, Vietnam

    Quy-Cao Tran

  3. School of Mechanical Engineering, Hanoi University of Science and Technology, No. 1 Dai Co Viet Street, Hanoi, Vietnam

    Quy-Cao Tran & Hai-Le Bui

  4. International Institute for Computational Science and Engineering, Hanoi University of Science and Technology, No. 1 Dai Co Viet Street, Hanoi, Vietnam

    Hai-Le Bui

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  1. Van-Binh Bui
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Correspondence to Hai-Le Bui.

Appendix

Appendix

In this section, designing of the conventional fuzzy controller (FC) which is similar to the HAC is presented. Control diagram of the analogical FC with two-input state variables \(x_{1}\) and \(\dot{x}_1\) and one-output control variable u with actuator saturation is shown in Fig. 27, in which \(\hbox {sat}(u)\) is expressed in (2). Rule base of the FC is presented in Table 6 and fuzzifications of the linguistic variables are shown in Figs. 28, 29 and 30, where Z is stood for linguistic value “Zero”. Mamdani method and centre gravity method are used as inference engine and de-fuzzification method of the FC.

Fig. 27
figure 27

Control diagram of the FC with actuator saturation

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Table 6 Rule base of the FC
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Fig. 28
figure 28

Fuzzification of \(x_{1}\)

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Fig. 29
figure 29

Fuzzification of \(\dot{x}_1 \)

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Fig. 30
figure 30

Fuzzification of u

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Bui, VB., Tran, QC. & Bui, HL. Multi-objective optimal design of fuzzy controller for structural vibration control using Hedge-algebras approach. Artif Intell Rev 50, 569–595 (2018). https://doi.org/10.1007/s10462-017-9549-3

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