research article novel distributed pzt active vibration ...ectiveness of the proposed active...

Research ArticleNovel Distributed PZT Active Vibration Control Based onCharacteristic Model for the Space Frame Structure

Hua Zhong,1 Yong Wang,2 Hanzheng Ran,1 Qing Wang,1 and Changxing Shao2

1 Institute of Electronic Engineering, China Academy of Engineering Physics, Mianyang 621999, China2Department of Automation, University of Science and Technology of China, Hefei 230027, China

Correspondence should be addressed to Hua Zhong; [email protected]

Received 6 July 2015; Revised 19 September 2015; Accepted 21 September 2015

Academic Editor: Mario Terzo

Copyright © 2016 Hua Zhong 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.

A novel distributed PZT control strategy based on characteristic model is presented for space frame structure in this paper. It is achallenge to obtain the exact mechanical model for space structure, since it is a coupling MIMO plant with unknown parametersand disturbances.Thus the characteristic modeling theory is adopted to establish the needed model, which can accurately describethe dynamic characteristics of the space frame structure in real time. On basis of this model, a keep tracking controller is designedto suppress the vibration actively. It is shown that the proposed model-free method is very robust and easy to implement. To solvethe complex and difficulty problem on PZT location optimization, an efficient method with modal strain energy and maximumvibration amplitude is proposed. Finally, a simulation study is conducted to investigate the effectiveness of the proposed activevibration control scheme.

1. Introduction

Space structures are usually applied in the aerospace andcivil engineering such as the space boom structures [1].Generally, since space structures are light and flexible, theyare sensitive to vibration which will affect the performanceof space equipment [2]. Therefore, it is necessary to suppressvibration for space structure.

Active vibration control method is a good choice forthe vibration control of space structure. To achieve goodvibration control performance, there are three things to bedone, that is, system modelling, sensors/actuators locationconfiguration, and control strategy design.

Space structure’s model can be divided into two cat-egories: mechanical model and identification model. Themechanical model is usually established with sophisticatedfinite element method [3, 4]. Additionally, it is generally usedin system dynamics analysis and sensors/actuators locationoptimization. Nonetheless, it can not be used to designcontroller directly due to its complexity and inaccuracy. Dueto the tremendous efforts devoted by researchers, valuableresults on identificationmodeling for the space structure have

been obtained. Hwang presented an analytical procedurebased on the Kalman filtering approach to estimate modalloads applied on a structure [5]. Hazra et al. proposed amethod which integrates the blind identification with time-frequency decomposition of signals for flexible structures [6].Schoen et al. presented an identification algorithmwhich uti-lizes modal contribution coefficients to monitor the data col-lection for large flexible space structures [7]. Although someaforementionedmethods have been extensively used in spacestructure identification, there still exist several disadvantages.On the one hand, the system is inherently complicated, muchless the disturbances. On the other hand, the existing resultsonly depend on complex off-line identification methods,which is difficult to grasp the system dynamics in real time.Consequently, it is a challenge to establish themodel for spacestructure with on-line identification method.

For active vibration control of space structure, determina-tion of actuator locations is an important work. If locationsof actuators are not proper, active vibration control maynot suppress the vibration, even it can increase structurevibration. Therefore, optimization of actuators’ locations hasattracted many scholars to study. Chen et al. presented

Hindawi Publishing CorporationShock and VibrationVolume 2016, Article ID 5928270, 9 pageshttp://dx.doi.org/10.1155/2016/5928270

2 Shock and Vibration

a location optimization approach based on the 1st singularvalue perturbations of observability and controllability [8].Xu and Jiang demonstrated an approach with the control-lability and observability index of the system [4]. Zhao etal. proposed a continuous variable optimization method tosolve the optimal placement of piezoelectric active bars [2].Li and Huang presented a layered optimization strategy toaddress location optimization problem [9].Though the abovemethods can realize location optimization actuators, theoptimization process is complex. Thus, it is needed to find asimple and effective optimization method.

An active vibration controller with simple structure andstrong robustness is necessary. Many scholars have devotedthemselves to solving this problem. Luo et al. designed simplePD controller for vibration attenuation in Hoop truss struc-ture [10]. Abreu et al. used a standard 𝐻

∞robust controller

to suppress structural vibrations [11]. Yang et al. presentedan adaptive fuzzy sliding mode controller for vibration ofa flexible rectangular plate [12]. Lin and Zheng proposed aparallel neurofuzzy control with genetic algorithm tuning forsmart piezoelectric rotating truss structure [13]. de Abreu etal. designed a self-organizing fuzzy controller for vibrationcontrol of smart truss structure [1]. Mahmoodi developed amodified positive position feedback controller for vibrationcontrol [14]. Wilhelm and Rajamani applied LQR controllerto realize multimodal vibration suppression [15]. Amongabove control strategies, simple controllers (such as PID)are with weak robustness. The complex controllers (𝐻

∞

controller, fuzzy controller) have strong robustness, but theyhave difficulty in parameter tuning.

Motivated by the above discussions, a novel distributedPZT control based on characteristic model is presented.Characteristic model for space structure is established toaccurately describe system’s dynamics in time whose formis simple and characteristic parameters of it can contain theinformation of time delay or high order [16]. Additionally, asimple controller based on characteristic model called keeptracking controller with strong robustness is applied to realizevibration control. Furthermore, a novel actuators locationoptimization method with modal strain energy (MSE) andmaximum vibration amplitude is proposed.

The remainder of the paper is organized as follows.Section 2 establishes the characteristic model for space struc-ture. Section 3 presents the identification of characteristicparameters and control strategy. PZT location optimizationis presented in Section 4. Numerical simulation is shown inSection 5. Conclusion is shown in Section 6.

2. Characteristic Modeling for Space Structure

Thespace frame structure is shown in Figure 1which contains26 nodes and 63 beam elements. Nodes 2, 3, 7, and 8are constrained which are fixed with the floor. Structureexcitation direction is 𝑧 of node 10. Every node has 6 degreesof freedom (DOF).The totalDOF is 𝑛.The space structure has63 beam elements. Table 1 shows the connection informationamong nodes. Space structure consists of aluminum whosecross section is rectangular. Table 2 shows the top 10 naturalfrequencies.

X

Y

Z

1 2 3 4

5

678

9

10

11

12

13

14

15

1617

18

19

20

21

22

23

2425

26

Figure 1: The schematic diagram of frame structure.

According to the finite element formulation, the equationof motion for any structure is given by

MẌ + CẊ + KX = F, (1)

whereM ∈ R𝑛×𝑛 is mass matrix, C ∈ R𝑛×𝑛 is damping matrix,K ∈ R𝑛×𝑛 is stiffness matrix, F ∈ R𝑛×1 is load force vector,and X ∈ R𝑛×1, Ẋ ∈ R𝑛×1, and Ẍ ∈ R𝑛×1 are structuraldisplacement, velocity, and acceleration vectors, respectively.

With the knowledge of space frame, parameters M, C,and K are slow time-varying. Then, during short samplingtime Δ𝑡 (ms), it can be considered thatM, C, and K are time-invariant. Then the discrete form of motion equation (1) canbe described as

MX (𝑘 + 1) = (2M − CΔ𝑡 − KΔ𝑡2)X (𝑘)

+ (−M + CΔ𝑡)X (𝑘 − 1) + Δ𝑡2F (𝑘) .(2)

One has the upper equation (2) as

X (𝑘 + 1) = G1(𝑘)X (𝑘) + G

2(𝑘)X (𝑘 − 1)

+H1(𝑘)F (𝑘) ,

(3)

where G1(𝑘) = 2I

𝑛− M−1CΔ𝑡 − M−1KΔ𝑡2, G

2(𝑘) =

−I𝑛+ M−1CΔ𝑡, and H

1(𝑘) = Δ𝑡

2M−1. The expression (3)is called characteristic model; G

1(𝑘), G

2(𝑘), and H

1(𝑘) are

characteristic parameters.When Δ𝑡 is very small, the following equation holds:

G1(∞) = 2I

𝑛,

G2(∞) = −I

𝑛,

H1(∞) = 0

𝑛×𝑛.

(4)

Further one has the sum of characteristic parameters as

G1(∞) + G

2(∞) +H

1(∞) = I

𝑛. (5)

Shock and Vibration 3

Table 1: Beam element connection information.

Beam number Node number1 1-22 2-33 3-44 4-55 5-66 6-77 7-88 8-99 9-110 9-1011 1-1012 2-1013 10-814 10-715 10-316 3-717 3-518 5-719 1-2620 25-1121 11-222 24-1223 12-324 23-1325 13-1426 14-427 15-628 16-1529 22-1630 17-731 21-1732 18-833 20-1834 19-935 19-2636 26-2537 26-1138 1-1139 25-2440 25-1241 11-1242 11-343 24-2344 24-1345 12-1346 12-1447 3-1448 23-2249 13-16

Table 1: Continued.

Beam number Node number50 14-1551 21-2252 21-1653 17-1654 17-1555 7-1556 20-2157 20-1758 18-1759 18-760 19-2061 19-1862 9-1863 24-21

Assuming the output X(𝑘) loops are decoupled, thusG1(𝑘) and G

2(𝑘) are diagonal matrixes, which can be formu-

lated as

G1(𝑘) =

[[[[[[

[

𝑔11

(𝑘) 0 ⋅ ⋅ ⋅ 0

0 𝑔12

(𝑘) ⋅ ⋅ ⋅ 0

.

.

.... d

.

.

.

0 0 ⋅ ⋅ ⋅ 𝑔1𝑛

(𝑘)

]]]]]]

]

,

G2(𝑘) =

[[[[[[[[

[

𝑔21

(𝑘) 0 ⋅ ⋅ ⋅ 0

0 𝑔22

(𝑘) ⋅ ⋅ ⋅...

.

.

.... d 0

0 0 ⋅ ⋅ ⋅ 𝑔2𝑛

(𝑘)

]]]]]]]]

]

.

(6)

The characteristic parameterH1(𝑘) can be written as

H1(𝑘) =

[[[[[[

[

ℎ11

(𝑘) ℎ12

(𝑘) ⋅ ⋅ ⋅ ℎ1𝑛

(𝑘)

ℎ21

(𝑘) ℎ22

(𝑘) ⋅ ⋅ ⋅ ℎ2𝑛

(𝑘)

.

.

.... d

.

.

.

ℎ𝑛1

(𝑘) ℎ𝑛2

(𝑘) ⋅ ⋅ ⋅ ℎ𝑛𝑛

(𝑘)

]]]]]]

]

. (7)

According to the expression in (3) and (6)-(7), the 𝑖thcharacteristic model is

𝑥𝑖(𝑘 + 1) = 𝑔

1𝑖(𝑘) 𝑥𝑖(𝑘) + 𝑔

2𝑖(𝑘) 𝑥𝑖(𝑘 − 1)

+

𝑛

∑

𝑟=1

ℎ𝑖𝑟(𝑘) 𝐹𝑟(𝑘) .

(8)


Table 2: Top 10 natural frequencies.

Order 1 2 3 4 5 6 7 8 9 10Frequency (Hz) 10.03 15.51 18.02 22.82 25.85 27.72 29.35 31.45 42.22 42.61

PZT1

PZT2

PZT3 Al beam

Figure 2: The schematic diagram of PZT structure.

Similarly, one has the characters of characteristic param-eters in 𝑖th output loop as

𝑔1𝑖(∞) = 2,

𝑔2𝑖(∞) = −1,

ℎ𝑖𝑟(∞) = 0,

𝑔1𝑖(∞) + 𝑔

2𝑖(∞) +

𝑛

∑

𝑟=1

ℎ𝑖𝑟(∞) = 1.

(9)

Because PZT with small size and light weight does notaffect structure’s characteristics after pasting on the structuresurface, PZT is selected as sensor and actuator. Figure 2indicates the schematic diagram of PZT structure. It is shownthat a pair of PZT1 and PZT2 act as actuator and anotherPZT3 acts as sensor. Additionally, actuators far away PZT3have little function on PZT3. Therefore, (8) can be simplifiedas

𝑥𝑖(𝑘 + 1) = 𝑔

1𝑖(𝑘) 𝑥𝑖(𝑘) + 𝑔

2𝑖(𝑘) 𝑥𝑖(𝑘 − 1)

+ ℎ𝑖𝑖(𝑘) 𝐹𝑖(𝑘) = Φ

𝑇

(𝑘) 𝜗 (𝑘) ,

(10)

where Φ(𝑘) = [𝑥𝑖(𝑘) 𝑥

𝑖(𝑘 − 1) 𝐹

𝑖(𝑘)]𝑇 and 𝜗(𝑘) =

[𝑔1𝑖(𝑘) 𝑔

2𝑖(𝑘 − 1) ℎ

𝑖𝑖(𝑘)]𝑇.

3. Keep Tracking Control Law

With a proper parameter estimator, the characteristic param-eters can adaptively converge to true values. The generalmethods of parameters identification are least squaremethodand gradient method. Compared with the least squaremethod, gradientmethod has a smaller amount of calculationand it will not cause parameters to diverge with improperforgetting factors.Thus, the gradientmethod [16] will be usedto estimate the characteristic parameters. The form is

�̂� (𝑘) =

𝜆1Φ (𝑘 − 1) [𝑥

𝑖(𝑘) −Φ

𝑇

(𝑘 − 1) �̂� (𝑘 − 1)]

Φ𝑇

(𝑘 − 1)Φ (𝑘 − 1) + 𝜆2

+ �̂� (𝑘 − 1) .

(11)

The range of𝜆1and𝜆

2is 0 < 𝜆

1< 1, 0 < 𝜆

2< 4.The estimates

of 𝑔1𝑖(𝑘), 𝑔

2𝑖(𝑘), and ℎ

𝑖𝑖(𝑘) in the characteristic model are

�̂�𝑇

(𝑘) = [�̂�1𝑖(𝑘) �̂�

2𝑖(𝑘) ℎ̂

𝑖𝑖(𝑘)].

To ensure the output 𝑦(𝑘+1) equals the desired referenceinput 𝑟(𝑘 + 1), the need control input should satisfy thefollowing condition:

𝑢 (𝑘) =𝑟 (𝑘 + 1) − �̂�

1𝑖(𝑘) 𝑦 (𝑘) − �̂�

2𝑖(𝑘) 𝑦 (𝑘 − 1)

ℎ̂𝑖𝑖(𝑘)

. (12)

To avoid the fact of ℎ̂𝑖𝑖(0) = 0, a slack variable 𝜆 is

introduced, which could guarantee that the control input issmooth and easy to implement. Consider

𝑢 (𝑘) =𝑟 (𝑘 + 1) − �̂�

1𝑖(𝑘) 𝑦 (𝑘) − �̂�

2𝑖(𝑘) 𝑦 (𝑘 − 1)

𝜆 + ℎ̂𝑖𝑖(𝑘)

, (13)

where 𝜆 is a constant. This control law is called characteristicmodel-based keep tracking control (CMKTC) law [16, 17].With the control law, system can perfectly track the referenceinput.

4. PZT Location Optimization

Considering the complexity of traditional optimization algo-rithm,MSE andmaximum amplitude are applied to optimizelocations of PZT whose optimization process is simple.

Firstly beam elements of space structure are dividedinto three categories: underside beam elements 𝑁

𝑢=

{1, 2, . . . , 17, 18}, front and rear beam elements 𝑁𝑓𝑟

=

{19, 20, . . . , 33, 34, 36, 37, . . . , 46, 47, 51, 52, . . . , 61, 62}, andconnection beam elements between front and rear𝑁𝑐= {35, 48, 49, 50, 63}. Through beams’ MSE, the sensitive

beams are obtained. With 𝑖th natural frequency, the 𝑗thbeam’s MSE [18] can be described as

𝐸𝑖𝑗=

𝜙𝑇

𝑖K𝑗𝜙𝑖

𝜔2

𝑖

, (14)

where 𝜙𝑖is modal shape vector with 𝑖th natural frequency

which is the 𝑖th column of Vnorm. Vnorm = V(√V𝑇M𝑉)−1is normalized modal matrix, where 𝑉 is modal matrix. K

𝑗is

𝑗th beam stiffness matrix; 𝜔𝑖is 𝑖th vibration frequency.

Figures 3–6 show the first four order MSE. Fromthe figures, sensitive beam element sets are obtained:underside sensitive beam element set 𝑁

𝑠= {1, 4, 5, 9, 10, 11},

front and rear sensitive beam element set 𝑁𝑏

=

{19, 20, 23, 25, 28, 34, 36, 37}, and connection sensitivebeam element set 𝑁

ℎ= {35, 48, 49, 50, 63}. 𝑁

𝑠, 𝑁𝑏, and 𝑁

ℎ

are the subsets of𝑁𝑢, 𝑁𝑓𝑟, and𝑁

𝑐, respectively.

Additionally, it is needed to select the most sensitivebeam element from 𝑁

𝑠, 𝑁𝑏, and 𝑁

ℎaccording to maximum


Table 3: 𝑧-direction maximum amplitude of beams in𝑁𝑠.

Beam number 1 4 5 9 10 11Maximum amplitude (mm) 0.0005 0.0013 0.0015 0.1422 0.3996 0.3996

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 181

36 51 6219

48 49 50 6335

Beam number

Beam number

Beam number

0

1

2

MSE

0

1

2

MSE

0

1

2

MSE

×10−3

×10−5

×10−6

Figure 3: First-order MSE.

48 49 50 6335

36 51 6219

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 181

Beam number

Beam number

Beam number

0

0.005

0.01

MSE

0

0.5

1

MSE

0

1

2

MSE

×10−4

×10−5

Figure 4: Second-order MSE.

amplitude. Excitation node is selected as node 10 and exci-tation direction is 𝑧. Excitation force 𝑓

𝑒= 10 sin(2𝜋𝑓𝑡),

and excitation frequency 𝑓 is selected as the first naturalfrequency 10.03Hz. Through analysing vibration responsecurves of 𝑁

𝑠, it is concluded that 𝑧-direction vibration

amplitude is much larger than 𝑥 and 𝑦 which is considered.Similarly, analysing vibration response curves of𝑁

𝑏and𝑁

ℎ, it

is shown that 𝑦-direction vibration amplitude is much largerthan 𝑥 and 𝑦. Thus, it is only needed to compare 𝑧-directionof the maximum amplitude for beams in𝑁

𝑏and𝑁

ℎ.

From Table 3 which shows the 𝑧-direction maximumvibration amplitude of 𝑁

𝑠, 10th or 11th beam element is

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 181

36 51 6219

48 49 50 6335

Beam number

Beam number

Beam number

0

1

2

MSE

0

5

MSE

0

5

MSE

×10−3

×10−5

×10−5

Figure 5: Third-order MSE.

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 181

36 51 6219

48 49 50 6335

Beam number

Beam number

Beam number

0

0.01

0.02

MSE

0

5

MSE

0

0.5

1

MSE

×10−5

×10−5

Figure 6: Fourth-order MSE.

the most sensitive beam element. In addition, comparedwith the other location on the 10th or 11th beam element,location approaching node 10 has larger vibration amplitudewhich is appropriate location for pasting PZT. Similarly,Table 4 indicates that 19th or 37th beam element is the mostsensitive beam element in𝑁

𝑏while the 19th beam element is

selected. Furthermore, considering that 𝑥-direction vibrationamplitude nearing node 19 is larger than the other positionon 19th beam element while 𝑦-direction vibration amplitudeon any location of 19th beam element is nearly the same,so position approaching node 19 on 19th beam element isanother appropriate location for pasting PZT. Lastly, 35th


Table 4: 𝑦-direction maximum amplitude of beams in𝑁𝑏.

Beam number 19 20 23 25 28 34 36 37Amplitude (mm) 0.1909 0.0930 0.0423 0.0713 0.0713 0.1909 0.0930 0.1909

Table 5: 𝑦-direction maximum amplitude of beams in𝑁ℎ.

Beam number 35 48 49 50 63Maximum amplitude (mm) 0.1909 0.1000 0.0703 0.0135 0.0714

Table 6: Maximum amplitude on 35th beam element.

𝑠 (m) 𝑥 (mm) 𝑦 (mm) 𝑧 (mm)0 0.00022 0.19 0.000480.2 0.02 0.19 0.0180.4 0.003 0.19 0.0250.6 0.018 0.19 0.0180.8 0.00018 0.19 0.00038

beam element is the most sensitive beam element in𝑁ℎfrom

Table 5. Since any location’s 𝑦-direction vibration amplitudeof 35th beam element has nearly the same value, 𝑧-directionvibration amplitude is considered.Define variable 𝑠 as the dis-tance between any location on 35th beam element and node19. Table 6 shows that themiddle location with themaximumvibration amplitude is another appropriate location to pastePZT.

In a word, three optimization positions are middle posi-tion on 35th beam element (1st controller CMKTC1), locationapproaching node 19 on 19th beam element (2nd controllerCMKTC2), and location approaching node 10 on 10th beamelement (3rd controller CMKTC3).

5. Numerical Simulation

5.1. Characteristic Model Verification Simulation. To verifythat characteristic model can accurately describe dynamiccharacters of system, with the literature [16], four forms ofcontrol input are selected as follows:

(a) Step signal 𝐹𝑖(𝑘) = 10.

(b) Smooth step signal 𝐹𝑖(𝑘) = 0.97𝑢(𝑘 − 1) + 0.3.

(c) 10Hz sinusoidal signal 𝐹𝑖(𝑘) = 10 sin(20𝑘𝜋Δ𝑡).

(d) 10Hz square wave signal 𝐹𝑖(𝑘) =

10 sign(sin(20𝑘𝜋Δ𝑡)).

Four different control input signals are shown in Figure 7.Control input direction is 𝑧-direction of node 10; vibrationdisplacement output direction is selected as 𝑦-direction ofnode 19. Output estimation error �̂�(𝑘) can be expressed as�̂�(𝑘) = 𝑥

𝑖(𝑘) − Φ

𝑇

(𝑘 − 1)�̂�(𝑘). Sampling time Δ𝑡 is selectedas 0.001 s.

Figures 8–11 show the output estimation error withdifferent control input. When control input is step signalor smooth step signal, output estimation error’s magnitudeis 10−7. If control input is sinusoidal signal or square wavesignal, output estimation error’smagnitude is 10−5.Therefore,

0

2

4

6

8

10

(a)(b)

(c)(d)

−2

−4

−6

−8

−10

Fi(k)

0.1 0.2 0.3 0.4 0.50Time (s)

Figure 7: Four-control input signal.

×10−7

0.2 0.4 0.6 0.8 10

Time (s)

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

ê(k)

Figure 8: Output estimation error with (a).

output of characteristic model can approach the outputof space structure. In addition, characteristic model canaccurately describe space structure’s dynamic characters.

5.2. Distributed PZT Vibration Simulation. Table 7 showsthe related parameters of PZT and Al. With charactersof characteristic parameters (9), select the initial value of


Table 7: Related parameters of PZT and aluminum.

Thickness (mm) Width (mm) Elastic modulus (Gpa) PSC (C/N)PZT 1 10 8 × 1010 5 × 10−10

Al 5 12 7 × 1010 —Remark 1: PSC: Piezoelectric Strain Constant.

Table 8: Control parameter.

Parameter CMKTC1 CMKTC2 CMKTC3𝜆 −0.0005 −0.0005 −0.025

×10−7

−1.5

−1

−0.5

0

0.5

1

1.5

ê(k)

0.2 0.4 0.6 0.8 10

Time (s)

Figure 9: Output estimation error with (b).

×10−5

−6

−4

−2

0

2

4

6

8

ê(k)

0.2 0.4 0.6 0.8 10

Time (s)

Figure 10: Output estimation error with (c).

characteristic parameters as �̂�𝑇

(0) = [1.618 − 0.618 0.03].Related parameters of gradient method are 𝜆

1= 0.8,

𝜆2

= 0.4. Sampling time Δ𝑡 is selected as 0.001 s. Controlparameters are shown in Table 8. Excitation force 𝑓

𝑒=

10 sin(2𝜋𝑓𝑡); excitation frequency 𝑓 is selected as the firstnatural frequency 10.03Hz. Select nodes 19, 22, and 25 asobservation nodes. It should be emphasized that the outputfeedback of CMKTC1 is 𝑦-direction displacement.

×10−5

ê(k)

1.5

1

0.5

0

−0.5

−1

−1.5

Time (s)10.80.60.40.20

Figure 11: Output estimation error with (d).

×10−4

×10−3

Node 19

0.1 0.2 0.3 0.4 0.50

Time (s)

0.1 0.2 0.3 0.4 0.50

Time (s)

0.1 0.2 0.3 0.4 0.50

Time (s)

Without controlCMKTC

−1

0

1

z(m

m)

−0.2

0

0.2

y(m

m)

−5

0

5

x(m

m)

Figure 12: Displacement of node 19.

Figures 12–14 are the vibration displacement curves ofnodes 19, 22, and 25, respectively. It is shown that 𝑦-directionvibration displacement is much larger than 𝑥 or 𝑦 andsuppressing vibration effect of 𝑦-direction is obvious. WithCMKTC, 𝑦-direction vibration displacement of nodes 19, 22,and 25 attenuates to 4 × 10−3, 4.5 × 10−4, and 1.8 × 10−4,respectively, which indicates the effectiveness of distributedPZT vibration control. Control input is presented in Figure 15which shows that control voltage is within voltage range ofPZT.


Table 9: 𝑦-direction vibration attenuation ratio of nodes 19, 22, and 25 with different amplitude sinusoidal disturbance.

Disturbance amplitude 10 20 30 40 50 60 70 80 90 100Node 19 (%) 96.91 96.52 96.07 95.65 95.18 94.74 94.28 93.82 93.35 92.72Node 22 (%) 96.61 96.71 96.42 96.16 95.90 95.45 95.01 94.54 94.10 93.65Node 25 (%) 96.36 95.27 96.47 93.73 92.00 90.33 88.60 86.90 85.20 83.50

Table 10: 𝑦-direction vibration attenuation ratio of nodes 19, 22, and 25 with different power-band white noise.

Noise power 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50Node 19 (%) 95.45 94.76 93.82 93.51 92.98 92.50 92.15 91.83 91.37 91.20Node 22 (%) 96.24 96.34 96.21 96.21 96.21 96.21 96.19 96.18 96.16 96.15Node 25 (%) 96.00 95.83 95.73 95.63 95.53 95.43 95.33 95.30 95.23 95.17

×10−3

×10−4

Node 22


−1

0

1

x(m

m)

−0.05

0

0.05

y(m

m)

−5

0

5

z(m

m)

0.1 0.2 0.3 0.4 0.50

Time (s)

0.1 0.2 0.3 0.4 0.50

Time (s)

0.1 0.2 0.3 0.4 0.50

Time (s)


×10−4

×10−5

Node 25


0.1 0.2 0.3 0.4 0.50

Time (s)

0.1 0.2 0.3 0.4 0.50

Time (s)

0.1 0.2 0.3 0.4 0.50

Time (s)

−5

0

5

z(m

m)

−0.05

0

0.05

y(m

m)

−5

0

5

x(m

m)


u1

(V)

0.1 0.2 0.3 0.4 0.50

Time (s)

0.1 0.2 0.3 0.4 0.50

Time (s)

0.1 0.2 0.3 0.4 0.50

Time (s)

−200

0

200

u3

(V)

−200

0

200u2

(V)

−200

0

200

Figure 15: Control input.

To verify the robustness of CMKTC, the matched distur-bance is added. When the sinusoidal matched disturbance𝑓𝑒= 𝐴 sin(200𝑡) with different amplitude is added, Table 9

shows 𝑦-direction vibration attenuation ratio of nodes 19,22, and 25. From the table, it is known that node 25is more sensitive than node 19 and node 22. No matterhow, sinusoidal disturbance has small effect on 𝑦-directionvibration displacement.

Considering that there exists noise in actual system, soband-white noise with different noise power is added. 𝑦-direction vibration attenuation ratio of nodes 19, 22, and 25is presented in Table 10. The table indicates that vibrationdamping ratio decreases with the increase of noise power.Of course, band-white noise has small effect on systemperformance and CMKTC is with strong robustness.

6. Conclusion

In this paper, a novel distributed PZT control strategybased on characteristic model has been proposed. To avoidthe difficulty in obtaining the mechanical model of spacestructure, an efficient model-free modeling approach, thatis, the so-called characteristic model theory, is adopted.


Then a PZT position optimization method based on MSEand maximum vibration amplitude has been presented, inwhich the placement optimization of sensors and actuatorsis realized. Due to the established characteristic model, asimple but efficient CMKTC is designed. The power ofthe proposed control strategy is demonstrated on severalnumerical examples.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgment

This work was supported by the National Natural ScienceFoundation of China (no. 61573332).

References

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