Placement optimization of actuator and sensor and decentralized

SCIENCE CHINA Technological Sciences • RESEARCH PAPER •

April 2011 Vol.54 No.4: 853–861 doi: 10.1007/s11431-011-4333-0

Placement optimization of actuator and sensor and decentralized adaptive fuzzy vibration control for large space intelligent truss structure LI DongXu*, LIU Wang, JIANG JianPing & XU Rui College of Aerospace and Material Engineering, National University of Defense Technology, Changsha 410073, China Received December 8, 2010; accepted January 31, 2011; published online March 1, 2011

Large space truss structure is widely used in spacecrafts. The vibration of this kind of structure will cause some serious problems. For instance, it will disturb the work of the payloads which are supported on the truss, even worse, it will deactivate the spacecrafts. Therefore, it is highly in need of executing vibration control for large space truss structure. Large space intelligent truss system (LSITS) is not a normal truss structure but a complex truss system consisting of common rods and active rods, and there are at least one actuator and one sensor in each active rod. One of the key points in the vibration control for LSITS is the location assignment of actuators and sensors. The positions of actuators and sensors will directly determine the properties of the control system, such as stability, controllability, observability, etc. In this paper, placement optimization of actuators and sensors (POAS) and decentralized adaptive fuzzy control methods are presented to solve the vibration control problem. The electro-mechanical coupled equations of the active rod are established, and the optimization criterion which does not depend upon control methods is proposed. The optimal positions of actuators and sensors in LSITS are obtained by using genetic algorithm (GA). Furthermore, the decentralized adaptive fuzzy vibration controller is designed to control LSITS. The LSITS dynamic equations with considering those remaining modes are derived. The adaptive fuzzy control scheme is improved via sliding control method. One T-typed truss structure is taken as an example and a demonstration experiment is carried out. The experimental results show that the GA is reliable and valid for placement optimization of actuators and sensors, and the adaptive fuzzy controller can effectively suppress the vibration of LSITS without control spillovers and observation spillovers. large space intelligent truss system, placement optimization of actuators and sensors, genetic algorithm, adaptive fuzzy control, decentralized control Citation:

Li D X, Liu W, Jiang J P, et al. Placement optimization of actuator and sensor and decentralized adaptive fuzzy vibration control for large space intelligent truss structure. Sci China Tech Sci, 2011, 54: 853861, doi: 10.1007/s11431-011-4333-0

1 Introduction Large space truss structure (LSTS) is widely used in spacecrafts, such as the supporting truss structure in shuttle radar topography mission and the supporting truss structure in space station [1]. This type of structure is of some special dynamical properties, say, low stiffness, low damping ratios, and low *Corresponding author (email: [email protected]) © Science China Press and Springer-Verlag Berlin Heidelberg 2011

natural frequencies. Due to these properties, the vibration of LSTS will happen very easily. If the truss structure is impacted by some particles from out space, or, if the spacecraft make a attitude maneuver, the structural vibration will be stirred up. As the vibration happens, it will cause many very serious problems to the payloads which are supported on the truss structure [2]. Hence, the issue on vibration control for LSTS is very important. In recent years, active vibration control of intelligent truss structure has achieved great progress by using piezoetech.scichina.com

www.springerlink.com

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lectric actuator and sensor [3–6]. Large space intelligent truss system (LSITS) is not a normal LSTS, but a complex truss system consisting of common rods and active rods. In each active rod, there are at least one actuator and one sensor. One of the key points in the vibration control for LSITS is the location assignment of actuators and sensors [6]. The placement optimization of actuators and sensors (POAS) is essential for the vibration control of LSITS. Unfortunately, the conventional method used for POAS is extremely time-consuming, which makes it unsuitable for a complex system such as LSITS. There is an urgent need to develop a new optimization method for POAS of this kind of large space intelligent truss structures. Genetic algorithm (GA) is a type of stochastic search algorithm that does not use an objective gradient function. This algorithm has fast convergence rate, by using it we can obtain a global optimized solution. Therefore, GA is feasible for POAS of LSITS. LSITS is a complex system. The space environment is uncertain. The dynamic process is nonlinear and time variable. Thus, it is difficult to set up a precise dynamic model of LSITS. Hence, those traditional control methodologies that are based on accurate dynamic model are not suitable for LSITS. Fuzzy control methodologies are robust and do not need a particularly precise dynamic model [7–9]. Marinaki et al. applied a fuzzy controller optimized by PSO for the vibration suppression of beams [10]. Sun used a generalized fuzzy predictive controller for the vibration suppression of flexible spacecrafts [11]. Zhu et al. used the fuzzy sliding control for the active vibration suppression during the attitude maneuver of spacecrafts [12]. All these studies obtained good control results. However, there is still very little work on space truss structures fuzzy vibration control. In this paper, one so called decentralized adaptive fuzzy vibration control method for LSITS is presented. There are several active rods with PZT stack actuator and PVDF (polyvinylidene fluoride) sensor in the LSITS. One active rod has one actuator-sensor pair. As the LSITS is a multi-input and multi-output (MIMO) system, if the general fuzzy control methods are put into use for a MIMO system such as LSITS, that will make the system to be too huge and too complex. That means it is nearly impossible to solve the vibration control problem of LSITS. The decentralized adaptive fuzzy vibration controller is designed to solve the vibration control problem of LSITS. The demonstration experimental results show that the GA is reliable and valid for placement optimization of actuators and sensors, and the adaptive fuzzy controller can effectively suppress the vibration of the LSITS without control spillovers and observation spillovers.

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piezoelectric material. Using the finite element method (FEM), the dynamic equations of the LSITS are developed via Lagrange equation. 2.1 Electro-mechanical coupled model of actuator and sensor The piezoelectric actuator under study is shown in Figure 1. When an external voltage is applied to the piezoelectric stack, it will provide the axial force. The spherical hinge can remove bending load. The spring provides preload and pre-displacement for the actuator. The piezoelectric actuator is modeled as a two DOF beam element, as shown in Figure 2. Based on the Lagrange equation, we can obtain a finite element dynamic equation of piezoelectric actuator, that is, e  e e M uu   K uu   F e  K u V,

e e e where M uu , K uu and K u are the element mass matrix,

element stiffness matrix and element coupled stiffness matrix, respectively;  is the generalized displacement vector, Fe is the generalized force vector, and V is the actuating voltage vector. If the PVDF sensor is embedded into the truss element with piezoelectric actuator, the equation of sensor can be approximately written as e Q e  kQu I Qu ,

(2)

e where Qe is the output vector of element charge, I Qu =[-1 0

0 1 0 0], kQu is a coefficient that can be obtained by experiment or derived from a specific sensor model. 2.2 Dynamic model of the LSITS A T-shaped intelligent truss structure with piezoelectric active rods is considered, as shown in Figure 3.

Figure 1

A piezoelectric actuator structure.

2 Modeling and formulation In this section, an LSITS with piezoelectric actuator is considered. The electro-mechanical coupling model of actuator and sensor are generated by using the actuating theory of

(1)

Figure 2

A piezoelectric actuator model.

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system; f mc  cT Bf a is the nc×1 dimensional modal control force vector. The matrix eq. (6) consists of nc uncoupled modal equations. Hence, we can design controller for each modal equation independently. The matrix cT B will be nonsingular through actuator placement optimization to avoid modal node. If modal control force vector fmc is known, the control force vector can be computed by



f a  cT B Figure 3

f mc .

(7)

V   Kv 

1

fa .

(8)

The matrix KlDc will be nonsingular through actuator position optimization. Therefore, the modal coordinate vector can be obtained as follows:

c   K l Dc  Q . 1

 M   C  K   f d  Bf a ,  Q  K l D 

(3)

where  Rn1 is nodal displacement vector, M Rnn, C Rnn, K Rnn are the mass matrix, damping matrix and stiffness matrix, respectively. fdRn1 is the disturbance force vector, B  R nnc , D  R ns n are the position matrix of actuators and sensors, respectively. Q  R ns 1 is the

 

output vector of the observer, K l  diag kli , kli  lsi bi ei

is the coefficient of the ith sensor, f a  R nc 1 is the control force induced by actuators, and fa  K v V ,

 

1

Then, the control voltage vector can be obtained from

T-shaped model of LSITS.

The intelligent space truss is composed of aluminum rods and PZT active rods. Assuming that the freedom of the whole system is n, and there are nc PZT actuators and ns PVDF sensors. The finite element dynamic equations of the system are

e where K v  diag kuv i



(9)

Based on the analysis above, a vibration control experimental system of the intelligent space truss structures is developed, as shown in Figure 4. The controller (shown in Figure 4) is designed to control the first nc modes. However, these controlled modes will be interacted with the residual modes, we need to further consider this issue. If all modes are considered, the output vector of the observer is

c   K l Dc  K l D 1

 c   K l Dc  K l Drr , 1

(10)

(4)

where r  R n( n - nc ) is the residual modal vectors matrix,

is a diagonal matrix, kuve i is the

  R n1 is the modal coordinate vector, and r  R ( n  nc )1 is the residual modal coordinate vector. If all the modes are considered, the modal control force vector will be

control force induced by the i th actuator under unite voltage, and V is the control voltage vector. The physical coordinates  can be represented by the modal coordinates:

  cc ,

(5)

where c is the modal matrix composed of the first nc modal vectors, and c is the corresponding modal coordinates. Substituting eq. (5) into eq. (3) and then left multiplying eq. (3) by vector cT , we can obtain  Mc  Cc  Kc  cT f d  f mc ,  Q  K l Dcc ,

f mc  f mc   Τ Bf a   T r B cT B





1

 . f mc  

Eq. (11) shows that the output of the controller will affect the residual modes. That is the reason causing controller

(6)

where M = cT M  diag 1 , C = cT C  diag  2ii  ,

K = cT K  diag i2  , i  1, 2, , nc ; i , i are the ith natural frequency and damping ratio of the structural

(11)

Figure 4

Intelligent space truss IMSC system.

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spillover and observation spillover. Eq. (10) can be rewritten as

c  c   K l Dc  K l Drr . 1

a a min V TV  f c T f c  max V TV .

(12)

Substituting eq. (11) into eq. (6), the modal equations considering the residual modes can be obtained as follows i ci  2iici  i 2ci  gi  f mc (i  1, 2,  , nc ) ,

Eq. (16) indicates that the actuating efficiency is closely related to the eigenvalues of matrix A . If all of the eigenvalues are concentrated and large, it can be ensured to get high actuating efficiency. The optimization criteria is chosen as

(13)

 nc  J a     aj  nc  j 1 

where gi is the ith row of matrix g, g   K l Dc  K l Dr  Mr  Cr  Kr  , 1

In the dynamic eq. (13), the residual modes are considered as a disturbance term. For the controlled system (13), the traditional fuzzy control methodologies in which control law is fixed will result in control spillover and observation spillover. However, the adaptive fuzzy controller, which can adjust control laws adaptively, will avoid this trouble. The adaptive fuzzy controller is obtained in section 4.

3 Placement optimization of actuators and sensors The POAS becomes an increasingly important issue in smart structure area. The POAS can not only affect the control efficiency and the energy consumption of the system, but also strongly affect the stability and reliability of the system. The POAS includes two aspects: the location and the number of actuators and sensors. In this section, an optimization criterion that does not depend upon the control method is proposed. Optimization criterion

The purpose is find the optimal placement in which the maximum modal control force and the maximum charge output are obtained for actuator and sensor respecting. Let A  cT K u ; the relationship between control voltage and modal control force can be written as V AV  f c f c , T

T

(14)

(15)

a a where min and max are the minimum and the maximum

eigenvalues of matrix A , respectively. Substituting eq. (14) into eq. (15), one can obtain

j 1

a j

)  max ,

(17)

Similarly, the optimization criteria of sensors can be chosen as  nc  J s     sj  nc  j 1 

nc

 ( j 1

s j

)  max ,

(18)

where  sj are the eigenvalues of matrix B T B , while B  K Quc .

In this paper, the PVDF sensor is on the tip of the PZT stack, which means that the actuators and sensors should appear in pairs. Therefore, optimization criteria can be chosen as J  J a  J s  max .

3.2

(19)

Application of GA

GA is a global search algorithm based on the fittest survival ideas of Darwin’s evolution theory, which uses genetic operations of reproduction, crossover, mutation, etc. to simulate the natural evolution process. In order to solve the optimization problem, the binary-coded GA is used for POAS of LSITS in this paper. 1) Encoding and decoding. If there are n rods in the truss, each rod can be given one serial number from 1 to n. And the serial number can be used as the optimization variable, which ranges from 1 to n. The optimization variable is encoded to a binary string. Let x be the optimization variable, and m be the length of the string; then 2m 1  n  1  2m  1 .

where A = AT A is an nc×nc positive definite symmetric matrix. The following inequality is hold a a min V TV  V T AV  max V TV ,

nc

 (

where  aj are the eigenvalues of matrix A .

i and f mc is the output vector of the controller.

3.1

(16)

(20)

The optimization variable x can be decoded using the following formulation x  1  decical ( y ) 

n 1 , 2m  1

(21)

where decical ( y ) denotes the decimal value of binary string y, and    stands for the rounding function. 2) Evaluating the fitness value. Since the actuators will affect the dynamic characteristics of the entire truss system, if the location of any actuator changes, the parameters of LSITS should be recalculated.

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In order to avoid repetitive counting, the following procedure is proposed. Firstly, the mass and stiffness matrix of truss structure without actuators are calculated. Secondly, the fitness value of an actuator located at any place is calculated. Thirdly, the mass matrix and stiffness matrix of actuators are assembled. Fourthly, one can calculate the modal eigenvectors, then conduct the modal coordinate transformation. Finally, the fitness value will be calculated via eq. (17). Generally, the sensors have little dynamic effects on the system, and we can ignore the sensor effects in the placement optimization of sensors. 3) Crossover. Let the crossover probability be pc, and the initial population size be pop; then select out n pairs of chromosomes, where n=, as the parents randomly, and then generate the new offspring via the multiple breakpoints crossover method. This method divides the chromosomes into multiple sub-strings according to the actual situation. The crossover process will generate a breakpoint from each sub-string randomly. New offspring will be produced through exchange the genes at the right of the parents’ breakpoints. 4) Mutation. Suppose that the mutation probability is pm. A uniform distribution number c between [0, 1] is generated for each gene randomly. If c≤pm, then the gene makes negation operations, for example, if the original value is 0, then it becomes 1 after the mutation. 5) Selection. The selection operation in this paper is based on the choice of an expanded sample space. According to this selection strategy, μ parents have to compete and survive with their λ offspring through the operations of crossover and mutation. Finally, the maximum fitness value is selected out for the next generation of parents. The process of GA optimization is illustrated in Figure 5.

4 Controller design There are three steps in fuzzy controller design: 1) Choosing the fuzzy controller inputs and outputs; 2) designing each of the components of the fuzzy controller; 3) choosing the pre-processing needed in the controller inputs and the post-processing needed in the outputs. In this paper, the outputs of the fuzzy controllers are the control voltages of the active rods. 4.1

Input variable choosing

The output signal of the PVDF sensor is an internal force of the rod along the axial line of the rod. It is expressed as follows:

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Figure 5

The process of GA optimization.

fink  ke e k  ke d33V ,

(22)

where ke  k1k2 k3 /(k1k2  k2 k3  k3 k1 ) is the equivalent stiffness of the active rod. k2  E2 A2 / l2 is the equivalent stiffness of the PZT stack. k1 and k3 are the equivalent stiffnesses on both ends of the PZT stack.  k is the displacement vector between both nodes of the active rod in the rod coordinate system. d33  nd33 is the equivalent piezoelectric constant of the PZT stack in the z-direction. e = [1 0 0 1 0 0], V is the control voltage. Since fink contains only actuating force item ke d33V , fink cannot reflect the system state exactly. It may induce

phoniness control (when fink  0 , the space truss vibration may still exist). Thus, fink should not be selected as fuzzy control input signal directly. Eq. (22) can be transformed as follows:

 k  e k  fink / ke  d33V ,

(23)

where  k is the node displacement difference of the kth active PZT rod, which can reflect the system state exactly. Hence, it is better to choose  k as fuzzy controller input signal. For every 2-dimentional fuzzy controller, the input signals are the error ek and the error change rate ck . They can be expressed as follows:

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ek (nT )   k (nT ),  ck  k  ( k (nT )   k (nT  T )) / T ,

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(24)

where T stands for a sampling period. 4.1 Member function and rule base

Seven fuzzy sets (NL, NM, NS, ZO, PS, PM, PL), two input power sets (E, C), and one output power set (V) are used in our research. Labels NL, NM, NS, ZO, PS, PM, PL refer to the linguistic values: Negative large, negative medium, negative small, zero, and positive small, positive medium and positive large, respectively. The triangular membership function is applied to the fuzzy sets, as shown in Figure 6. The rule base used in this paper is shown in Table 1. The first row and the first column of this table are the inputs to the system while others are the outputs of the fuzzy logic controller. Knowledge about the vibrating LSITS is employed to construct this rule base. The control output surface of LSITS is shown in Figure 7. 4.2

Adaptive law

Figure 7

Control surface.

where ci is the ith row of the output vector of the observer. The adaptive fuzzy controller in ref. [13] uses the state variable x as input, and its output is u D ( x /  )    ( x ) ,

(26)

where   R m1 is the parameter vector of the control rules, which is changeable,  ( x ) is an m-dimensional vector，the kth element is n

The decoupled eq. (13) does not consider the disturbance of residual modes. Hence, we can design an adaptive fuzzy controller for every equation independently. The output m  0 for every control quantity is ci . Choosing the error ei and the error change rate eci as inputs of the ith fuzzy controller, we can obtain

Table 1

(25)

Membership function plot.

Fuzzy IF-THEN rule base

u

e

e

NB NM NS ZO PS PM PB

k i

M

( xi )

n

(  A ( xi ))  l 1 i 1

NB

NM

NS

ZO

PS

PM

PB

PB PB PB PB PB PM PS

PB PB PM PM PS ZO NS

PB PM PS PS ZO NS NM

PB PM PS ZO NS NM NB

PM PS ZO NS NS NM NB

PS ZO NS NM NM NB NB

NS NM NB NB NB NB NB

.

(27)

l i

The adaptive law is

   pnT e ( x ) ,

ei  m  ci  ci ,  eci  ei  ci ,

Figure 6

k 

A  i 1

(28)

where  is the learning rate, pn is a coefficient vector, and e is a state error vector. There exist some disadvantages in the controller designed above: 1) Since the state variable x is used as input to the system, the controller may be worse and worse due to the fuzzy rules being modified by a wide margin when the ideal output changes intensively; 2) Only control performance pnT e is considered in modifying control rules; 3) There is no limitation to the control rules modification. Thus, the control rules will be modified repeatedly and exorbitantly even divergently when expected output changes intensively. In this paper, the improved methods are as follows. 1) The input variable of the fuzzy controller is changed as error variable e  (e, e,  e ,  , e( n 1) ) . By this way, the changed fuzzy rules will have inheritance even the expected output changes intensively. 2) Based on sliding control theory, if S  pnT e , as S  S   S , the system can transfer to the sliding surface

automatically. That means both S and S should be taken into account in modifying control rules.

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3) The upper and lower limits of  can be changed by real control state. The structure of the improved adaptive fuzzy controller is shown in Figure 8. 4.3

Scaling factors

Scaling factors are obtained from experience and system debugging. Their values are uncertain, but there are some characteristics that should be known. If the error factor Ke increases, the stiffness of the dynamic system increases. If the error change rate factor Kc increases, the damping coefficient increases. However, if Ke or Kc is too high, the controller will be too sensitive and cause energy waste. Since the control force is limited, the vibration of LSITS will be an oscillation attenuation process. Hence, error factors Ke should not be too big, otherwise, the stiffness of the system will increase and cause oscillation badly. However, error change rate factors should be bigger in order to increase damping ratio of the system. Generally, proportion factors Ku are decided by the maximum control voltage that the spacecraft can supply. That is, K u  Vmax / | I | ,

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5.1

Parameters

The LSITS with fuzzy vibration control system is shown in Figure 9. This truss is made from aluminum hollow rod, whose parameters are shown in Table 2. The construction of the active rods is shown in Figure 10, and their parameters are shown in Table 3. The first 6 modals are considered in the experimental.

(29)

where Vmax is the maximum control voltage that the spacecraft can supply, and |I| is the absolute maximum value of the control surface. Figure 9

5 Experimental Results

The experiment system.

In this section, a T-typed truss structure is taken as an example and the experiment system is set up. In the demonstration experiment, GA is used to calculate the optimal placement of the active rods, the convergence process and the optimal placement are obtained. The decentralized adaptive fuzzy controllers are designed to control vibrations of the LSITS.

Figure 10 Table 2

Figure 8

Adaptive fuzzy controller.

The structures of active rod.

Parameters of space truss structure

Item Size (m) Rod number Elastic modulus (GN/m2) Density (kg/m3) Inside diameter (m) Outside diameter (m) Equivalent cross area (m2) Connector weight (kg)

Value 0.4×0.4×4.8 161 72.7 3100 0.012 0.008 6.28×105 0.2

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Table 3

Parameters of actuator PZT rod

Item Density (kg/m3) Equivalent cross area (m2) Elastic modulus (Pa) PZT stack density (kg/m3) PZT stack equivalent cross area (m2) PZT elastic modulus (Pa) PZT stack length (m) Equivalent piezoelectric constant d33 (m/V) PZT patch number

5.2

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Value 8000 6.283×105 2.1×1011 7600 7.07×104 6.3×1010 0.16 5.6×108 140

POAS result

There are 5 PZT actuators and 5 sensors. The optimal positions of the active rods are shown in Table 4. Figure 11 shows the fitness convergence curves of 5 active rods when considering the impact of actuators and sensors on the overall system of LSITS. The label “Max”, “Min” and “Mean” in Figure 11 denote the maximum, minimum and average fitness values of each generation, respectively. It can be seen from Figure 11 that population fitness will converge at the 120th generation. By using the method presented in this paper, the time of calculation for the objective function is less than 104. GA is very suitable for solving the problem of POAS. The search process has a fast convergence rate to save a great amount of computing time. 5.3

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N and width 0.01 s, is applied on the tip of the truss. Both responses of the system with adaptive fuzzy control and without control are shown in Figure 12. Case 2. Periodic excitation response. In this experiment, an excitation F0sinωt is applied on the tip of the truss, where F0=1.414 N and ω=31 r/s. The responses are shown in Figure 13. The experimental results show that if there is no control to the system, the vibration amplitude will be higher and higher, since the frequency of the excitation is close to the natural frequency of the truss. Case 3. White noise excitation response. In this experiment, a white noise excitation is applied on the tip of the truss. The time-domain responses of the truss with the fuzzy controller and without controller are shown in Figure 14. All of the three experiments demonstrate that the decentralized adaptive fuzzy controller can suppress the vibration of LSITS effectively.

Figure 12

Vibration control results

Responses of impulse excitation.

Usually, there are three main excitation modes (impulse excitation, periodic excitation and white noise excitation) in the space. The control effects of the adaptive fuzzy controllers are evaluated by considering these three excitation modes. Case 1. Impulse excitation response. In this experiment, an impulse force, with amplitude 200 Table 4

The results of optimal active rods placement

Placement (26), (37), (48), (3337), (3640)

Fitness 6.8×109

Note：(ij) in the table is the node numbering of both ends of rods.

Figure 11

The convergent process of active rods optimization.

Figure 13

Figure 14

Responses of period excitation.

Responses of white noise excitation.

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6 Conclusions 2

In this paper, decentralized adaptive fuzzy vibration control of intelligent space truss is studied. The conclusions are as follows. 1) The GA is a very effective way to solve the POAS problem. The algorithm has fast convergence speed, which significantly reduces the computing time, as compared with exhaustive rules. 2) The fuzzy controller designed in this paper provides a method and research foundation for the vibration control of large flexible space truss structures. 3) If the expected output changes smoothly, an adaptive fuzzy controller is better than a normal fuzzy controller, especially for nonlinear and time variable system. When the expected output changes intensively, the improved adaptive fuzzy controller is better than both the general fuzzy controller and the adaptive fuzzy controller. 4) The improved fuzzy controller can suppress the vibrations excited by impulse excitation, periodic excitation and white noise excitation effectively. 5) There are no control spillovers and observation spillovers when using the adaptive fuzzy controller.

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This work was supported by the National Natural Science Foundation of China (Grant No. 10472006).

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Kopsaftopoulos F P, Fassois S D. Vibration based health monitoring for a lightweight truss structure: Experimental assessment of several

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