The integrated multiobjective optimal design method for structural active control system is put forward based on improved Pareto multiobjective genetic algorithm, through which the position of actuator is synchronously optimized with active controller. External excitation is simulated by stationary filtered white noise. The root-mean-square (RMS) of structural response and active control force can be achieved by solving Lyapunov equation in the state space. The design of active controller adopts linear quadratic regulator (LQR) control algorithm. Minimum ratio of the maximum RMS of controlled structural displacement divided by the maximum RMS of uncontrolled structural displacement and minimum ratio of the maximum RMS of controlled structural shear divided by the maximum RMS of uncontrolled structural shear, together with minimization of the sum of RMS of active control force, are used as the three objective functions of multiobjective optimization. The optimization process takes the impact of structure and excitation parameter on the optimized results. An eight-storey six-span plane steel frame was used as an emulational example to demonstrate the validity of this optimization method. Results show that the proposed integrated multiobjective optimal design method is simple, efficient, and practical with good universality.
In the research area of structural active control, optimal design of the control system, especially the position optimization of actuator, is always one of the hottest hot points of research. In nature, the position optimization of actuator is an issue of the optimization of discrete domain [
Considering the high cost of initial investment and operation of structural active control system [
This paper, based on improved Pareto multiobjective genetic algorithm, puts forward an active control integrated optimal design method. Compared with the method proposed in previous literatures, the method of optimization design presented in this paper, which can consider the internal relations between the control device’s position optimization and the controller itself, can obtain the optimal position of actuator under the condition of the random excitation and stochastic optimization criterion. The optimization process of the method proposed in this paper can be carried out in the frequency domain, so that the computational efficiency is very high, which is applicable to the complex plane model’s optimization of active control system, and can obtain the optimal layout position of actuator. In the meanwhile, this has important reference value for the practical application in engineering.
In order to improve computing efficiency, the optimization process is carried out in frequency domain, the random seismic excitation is simulated by stationary filtered white noise. The variance of structural response and active control force can be worked out by obtaining the solution of Lyapunov equation of the structure under stationary filtered white noise in the state space. The design of active controller adopts linear quadratic regulator (LQR) controlling algorithm. Minimization of the ratio between the maximum root-mean-square (RMS) of controlled structural displacement and the maximum RMS of uncontrolled structural displacement and minimization of the ration between the maximum RMS of controlled structural shear and the maximum RMS of uncontrolled structural shear, together with minimization of the sum of RMS of active control force, are used as the three objective functions of multiobjective optimization. At the end, this paper takes a plane frame structure as an example to certify the effectiveness of proposed active control integrated multiobjective optimization design method.
The structural dynamic equation of
The state equation of formula (
The Clough-Penzien spectrum [
The state equation of earthquake input-structural model can be obtained as follows by combining formulas (
The solution of differential equation (
The related functional matrix of structural response can be calculated through the formula above, and its expression is given as follows:
In the formula above, the variance matrix of input excitation is defined as follows:
If we plug the formula above into formula (
Provided that
Taking the derivative of time
Variance
The structural dynamic equation of
In this formula,
Optimal feedback gain matrix
The controller design and state estimation are treated separately in accordance with separation principle [
In this formula,
Figure
Block diagram of stationary filtered white noise-controlled structure-controller.
An extended state equation can be obtained through expanding controlled structure, LQR controller, and filtered white noise into a state equation, which is expressed as
In this formula,
Similarly, the variances of both the controlled structure response and the control force under stationary random excitation can be worked out. The peak control force needed in actual engineering application can be figured out conveniently through RMS and peak factor.
Considering that the maximum output force of actuator is limited by its physical conditions in actual engineering application [
In this formula,
The previous constraints can be treated with penalty function method, and the definition of optimized objective function is
In this paper, improved multiobjective genetic algorithm based on Pareto optimal solution [
The issues of multiobjective optimization above can be expressed as minimized objective function vector, which is expressed as
In this formula,
Define optimal solution
Pareto ordering of two objectives.
Given that for all the subobjectives, there should be at least one subobjective that makes
According to the definition of domination, it is easy to find out that the individuals on optimal boundary are not dominated. Provided that its boundary set sequence number is 1 (
In the event that sequence numbers of boundary set are equal, the pros and cons of individuals are represented by clustering distance. As illustrated in Figure
Reference [ Randomly generate initial parent population Integrate the two populations ( Empty If Calculate the clustering distance of individuals in Set Perform roulette selection, intersection, and variation operation on If
Flow diagram of improved NSGA-II algorithm.
This paper takes an eight-layer six-span frame structure as an example to examine the effectiveness of new integrated multiobjective optimal algorithm for active control system. As illustrated in Figure
Structural parameters.
Storey number | Column | Beam | ||
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Bending inertia moment |
Cross-sectional area |
Bending inertia moment |
Cross-sectional area |
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1-2 |
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3-4 |
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5-6 |
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7-8 |
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Finite element model of the structure installed with actuators.
A specific set of reference values of parameters
Design value of site soil parameters.
Parameter | Site classification | |||
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I | II | III | IV | |
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25.12 | 17.96 | 13.95 | 9.68 |
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0.65 | 0.71 | 0.82 | 0.91 |
Optimal frontier curve is illustrated in Figure
Optimal frontier curve of Pareto.
The actuators’ positions and relevant control gains corresponding to some optimal individuals selected from Pareto optimal frontier curve are listed in Table
Optimized results of site classification I when linear stiffness ratio of beam to column is 4.5.
Serial number | Position of the actuator |
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1 | 1, 2, 6, 7, 9, 12, 27, 33, 39, 43, 45, 48 | 0.23 | 0.37 | 481 |
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2 | 7, 12, 25, 36, 37, 42, 43, 44, 45, 46, 47, 48 | 0.48 | 0.54 | 193 |
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3 | 4, 12, 26, 35, 38, 41, 43, 44, 45, 46, 47, 48 | 0.69 | 0.73 | 92 |
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4 | 1, 2, 4, 5, 6, 7, 8, 9, 11, 12, 13, 15 | 0.26 | 0.36 | 513 |
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Optimized results of site classification III when linear stiffness ratio of beam to column is 4.5.
Serial number | Position of the actuator |
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1 | 3, 5, 7, 10, 21, 29, 33, 37, 41, 44, 46, 48 | 0.22 | 0.34 | 405 |
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2 | 12, 24, 34, 36, 38, 40, 42, 44, 45, 46, 47, 48 | 0.45 | 0.47 | 189 |
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3 | 11, 24, 35, 36, 37, 39, 42, 44, 45, 46, 47, 48 | 0.63 | 0.65 | 112 |
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4 | 1, 2, 4, 5, 6, 7, 8, 9, 11, 12, 13, 15 | 0.22 | 0.35 | 465 |
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Optimized results of site classification III when linear stiffness ratio of beam to column is 0.45.
Serial number | Position of the actuator |
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1 | 1, 2, 6, 7, 12, 27, 31, 36, 39, 43, 45, 48 | 0.27 | 0.35 | 312 |
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2 | 12, 24, 34, 36, 38, 40, 42, 44, 45, 46, 47, 48 | 0.45 | 0.52 | 155 |
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3 | 11, 24, 33, 36, 38, 39, 42, 44, 45, 46, 47, 48 | 0.64 | 0.70 | 82 |
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4 | 1, 2, 4, 5, 6, 7, 8, 9, 11, 12, 13, 15 | 0.28 | 0.37 | 339 |
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It turns out by the simulation analysis that the optimal controller of active control system is closely related to the position of actuator and that they both are influenced by structural parameters and excitation characteristics. That is why the design of traditional active control system could not obtain optimal seismic reduction effect, because the optimization is done separately into system design optimization and action controller position optimization. As a result, optimal control effect can be obtained only when an integrated design in structure-control system is carried out.
Integrated multiobjective optimal design for active control system is presented in this paper by using Pareto optimal solution-based improved multiobjective genetic algorithm and random vibration theory. The conclusion is given as follows. It is necessary for active control system to make integrated optimal design of the control device position and the controller. Previous optimal design is done separately into controller design optimization and action controller position optimization, which is the major reason why optimal seismic reduction effect cannot be achieved. The Pareto optimal solution set based on improved Pareto multiobjective genetic algorithm provides decision-makers with even wider choices, with which they may select corresponding layout schemes in accordance with actual projects. Therefore, it has high project application value.
With simplicity, efficiency, and practicability, integrated multiobjective optimization method for active control system put forward in this paper is expected to have promising engineering application.
The authors declare that there is no conflict of interests regarding the publication of this paper.
Financial supports for this research are provided by the National Natural Science Foundation of China (no. 51078077) and the National Science and Technology Pillar Program during the Twelfth Five-Year Plan Period (no. 2012BAJ14B00). These supports are gratefully acknowledged.