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Active vibration controller design and comparison study of flexible linkage mechanism systems
Mechanism and Machine Theory 37 (2002) 985–997 www.elsevier.com/locate/mechmt
Active vibration controller design and comparison study of flexible linkage mechanism systems Zhang Xianmin
a,*
, Shao Changjian a, Arthur G. Erdman
b
a
b
Department Mechatronic Engineering, Shantou University, Shantou, Guangdong 515063, PR China Department of Mechanical Engineering, University of Minnesota, 111 Church Street S.E., Minneapolis, MN 55455-0111, USA Received 23 March 2001; accepted 11 February 2002
Abstract Three active vibration controller design methodologies include the reduced modal controller, the classical and the robust H1 controller for the high-speed flexible linkage mechanism systems with piezoelectric actuators and sensors are investigated. Firstly, the state space form control model is formulated based on the complex mode theory. Secondly, the reduced modal controller, the classical and the robust H1 controller are designed. Finally, numerical simulations are carried out on a 4-bar linkage mechanism. The simulation results show that the vibration of the system is significantly suppressed with permitted actuator input voltages by each of the three controllers. The reduced modal control makes it more convenient to increase the modal damping of the system than that of the H1 control does, while the robust H1 control can avoid the spillover due to mode truncation. Ó 2002 Published by Elsevier Science Ltd. Keywords: Active vibration control; Flexible linkage mechanism; Modal; Controller; Actuator; Sensor
1. Introduction Recently, considerable attention has been paid to the investigation of the vibration control for high-speed flexible mechanism systems. Heretofore, there are basically five design philosophies, which have been developed to improve the elastodynamic responses for these kinds of systems. The first involves designing links of conventional materials with optimization of the cross-sectional geometry of the members [1,2]. The second one uses additional damping materials to
*
Corresponding author. Tel./fax: +86-754-2903313. E-mail address: [email protected] (Z. Xianmin).
0094-114X/02/$ - see front matter Ó 2002 Published by Elsevier Science Ltd. PII: S 0 0 9 4 - 1 1 4 X ( 0 2 ) 0 0 0 2 5 - 3
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Nomenclature system damping matrix system control matrix and output matrices system generalized force vector coefficient matrices output feedback robust controller transfer function identity matrix system stiffness matrix system mass matrix the number of the degrees of freedom of the system the number of the actuators and sensors bonded on an element the number of controlled and uncontrolled modes weighting matrices nodal displacement, velocity, and acceleration vectors of the whole linkage in the global frame u control input vector W weighting function suppositional input disturbance wW X, X0 the left and right eigenvector matrices x state space vector modal coordinate vector x0 D coordinate matrix b counteract coefficient e a given constant kiR , kiI par the real and image part of the ith complex eigenvalue C Da , Ds F F1 , F2 G H I K M n na , ns nc , nr Q, R q, q_ , € q
dissipate the vibration energy [3–5]. The third advocates that the mechanism links should be built of advanced composite materials because of their high damping and high stiffness-to-weight ratio [6,7]. These three design concepts may be referred to as passive vibration control. The fourth introduces a microprocessor-controlled actuator into the original mechanism to reduce the deflection of the flexible linkage [8–10]. The last involves the application of smart materials featuring distributed actuators and sensors to the linkage mechanisms to control unwanted vibration [11– 18]. The advantage of this method is that the controller can adapt with system changes, and thus can be much more effective than other methods. Sung and Chen [11] attempted to control the elastodynamic response of a four-bar linkage mechanism system with a flexible follower, a rigid crank, and a rigid coupler. Two patches of piezoceramic actuators were bonded to the surface of the follower at two locations that are symmetric with respect to its midpoint and one piezoceramic sensor was bonded to the surface at its midpoint. The authors assumed that the free vibration modes of the flexible follower link were the same as those of a simply supported beam. Optimal linear quadratic regulator (LQR) is employed in the control system. Liao and Sung [12] derived
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the finite element equations of the above mechanism and studied the active vibration control problem both analytically and experimentally based on the linear quadratic Gaussian theory. In the work of Ref. [13], Choi et al. investigated a slider-crank mechanism whose flexible connecting rod was bonded with two piezoelectric films to its upper and lower surface. One of the films was used as a distributed sensor while the other was used as a distributed actuator. The linear optimal feedback control law with a Luenberger observer was employed. On the basis of the independent modal control method, Zhang et al. [14,15] studied the active vibration control problem for the flexible mechanisms all of whose members were considered as flexible. By means of experimental studies, Thompson and Tao [16] controlled the connecting rod vibrations of a slider crank mechanism by using two sets of piezoelectric actuator/strain gauge sensor pairs. Classical position feedback controller was employed in the experimental system. Sannah and Smaili [17] presented an experimental investigation for controlling the elastodynamic response of a four-bar mechanism system with a flexible coupler link, slightly less flexible follower link, and relatively rigid crank. A controller, which consists of a LQR and a Luenberger observer, was designed and implemented. However, the state space matrices of the system were assumed to be constant for the entire motion cycle of the mechanism, which is not the case when the mechanism operates at high speeds. In Ref. [18], Shao and Zhang develop a hybrid independent modal controller which is composed of state feedback and disturbance feed-forward control laws. Like other methods, the control spillover problem may exist if the control is not imputed properly. In this paper, based on the complex mode theory and the control model developed in Ref. [19], the reduced modal controller, the classical H1 , and the robust H1 controller for active vibration control of the high-speed flexible linkage mechanism systems with piezoelectric actuators and sensors are investigated. Comparison study shows that the vibration of the system is significantly suppressed with permitted actuator voltages by each of these controllers. The reduced modal control makes it more convenient to increase the modal damping of the system than that of the H1 control does, while the robust H1 control avoids the spillover due to mode truncation. At the same level of control for vibration, the input voltage is higher and changes more violent than that of the robust H1 method. 2. Modelling of control According to the Hamilton theory, the mathematical model of control for the flexible linkage mechanism can be expressed as [19] M€ q þ Cq_ þ Kq ¼ F Da u
ð1Þ
y ¼ Ds q
ð2Þ
where M, C, K, Da , and Ds are the systematic mass, damping, stiffness, control and output matrices, respectively; F is the systematic generalized force vector; q, q_ , €q are the generalized displacement, the velocity, and acceleration vectors of the linkage system, respectively; u is the control input vector; and y is the output vector of the sensors. This model includes both the rigid body and the elastic motion coupling terms and the elastodynamics and piezoelectricity coupling terms, and takes into account the effects of the piezoelectric apparatus upon the mass and the stiffness of the system. It is noted that all the system matrices are not constant but periodically
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varying functions of the position of each links of the mechanism. Matrices C and K are not symmetric matrices, so the equations cannot be solved using the traditional real modal method. In order to decouple Eq. (1), we define a state space vector as x ¼ fq_ T ; qT gT ;
x_ ¼ f€ qT ; q_ T gT
Rewriting Eqs. (1) and (2) in terms of the state space vector yields Ax_ þ Bx ¼ w0 þ Du
ð3Þ
y ¼ Sx
ð4Þ
where
0 A¼ M
M ; C
M 0 B¼ ; 0 K
0 0 ; w0 ¼ D¼ F Da
S ¼ ½0
Ds
Based on the complex modal theory, one can obtain the state space form de-coupled equations of motion as [18] x_ 0 ¼ A0 x0 þ B10 w0 þ B20 u
ð5Þ
y ¼ C20 x0
ð6Þ
where
A0 ¼ Block diagð Aii Þ; C20 ¼ SX;
Aii ¼
kiR
kiI
kiI
kiR
B10 ¼ X0T ;
B20 ¼ X0T D;
i ¼ 1; 2; . . . ; n
where kiR and kiI i ¼ 1; 2; . . . ; n are the real and image part of the complex eigenvalues, n is the number of the generalized coordinates of the system, terms X and X0 indicate the normalized left and the right eigenvector matrices, respectively. From the control theory, one can consider the first number of nc vibration modes as controlled modes and leave a number of nr modes as uncontrolled modes with nc þ nr ¼ n. Let x0 ¼ fxTc
xTr gT
Eqs. (5) and (6) can be rewritten as B2c Ac 0 xc B1c x_ c ¼ þ w þ u xr B1r 0 B2r 0 Ar x_ r x y ¼ ½ C2c C2r c xr
ð7Þ ð8Þ
3. The reduced modal controller design The reduced modal control indicates that the controller is designed based on the controlled modes. Dividing the control input into two parts u ¼ u1 þ u2
ð9Þ
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where u1 is the state feedback term which allow to assignment the poles of the system to provide sufficient damping for the controlled modes, and u2 is the feed-forward terms which is used to counteract the disturbance input of the system. The two terms can be expressed as u1 ¼ F1 y;
u2 ¼ F2 w0
ð10Þ
where F1 , F2 are the state feedback and the disturbance feed-forward coefficient matrices, when we just consider the control modes, they satisfy following relations B2c F1 C2c ¼ DAc ;
B2c F2 ¼ bB1c
ð11Þ
where matrix DAc consists of the difference between the original and the required eigenvalues of the controlled modes, b is the counteract coefficient of the input disturbance. Properly selecting the positions of the actuators and the sensors, the matrices B2c and C2c can be full row ranked, then the first nc th modes are controllable and observable. Let the numbers of the actuators and the sensors be equal to nc . Solving the above equations one can obtain the state feedback and disturbance feed-forward coefficient matrices F1 , F2 as F1 ¼ ðBT2c B2c ÞT BT2c DAc CT2c ðC2c CT2c Þ1
ð12Þ
F2 ¼ bðBT2c B2c Þ1 BT2c B1c
ð13Þ
Substituting the two terms into Eqs. (7) and (8), considering the residual modes, one can further obtain the closed-loop equation as B2c F1 C2r Ac þ DAc xc ð1 bÞB1c x_ c ð14Þ ¼ þ w B2r F1 C2c Ar þ B2r F1 C2r xr B1r þ B2r F2 0 x_ r
4. Classical and robust H‘ controller design 4.1. The design performance index and the nominal control model The aim of this section is to design a controller u ¼ Gy
ð15Þ
which satisfies the following conditions (1) the closed-loop system is stable; (2) when the initial state vector xð0Þ ¼ 0, then at any time t the system satisfy the relation. Z t Z t T T J¼ ðx Qx þ u RuÞ dt < e wT0 w0 dt ð16Þ 0
0
where e is a given constant, Q and R are the weighting matrices. Definite the controlled output vector as z0 ¼ Nx þ Lu where Q ¼ NT N, R ¼ LT L.
ð17Þ
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When we just consider the controlled modes, the output vector can be expressed as z0 ¼ C1c xc þ D12c u where C1c ¼ NXc , D12c ¼ L, and Xc is the eigenvector matrix corresponding to the controlled modes. The equivalent form of Eq. (16) is Z t zT0 z0 dt J¼ 0
If we just consider the controlled modes in the performance index, then the nominal transfer function from the input vector to the output vector is
and the transfer function form of Eqs. (7) and (8) can be expressed as z0 P11 P12 w0 w0 ¼ ¼P y0 u P21 P22 u
ð18Þ
where P11 ¼ C1c ðsI Ac Þ1 B1c ; 1
P21 ¼ C2c ðsI Ac Þ B1c ;
P12 ¼ C1c ðsI Ac Þ1 B2c þ D12c 1
P22 ¼ C2c ðsI Ac Þ B2c
Then the equivalent form of Eq. (16) can be further expressed as pffiffi Tz w < c ¼ e 0 0 1
ð19Þ
where Tz0 w0 is the closed loop transfer function from w0 to z0 , k k1 indicates the H1 norm. Tz0 w0 ¼ P11 þ P12 GðI P22 GÞ1 P21 4.2. The unmodelled dynamics and the robust stability index The transfer function form of the full model (7–8) can be expressed as " 1 1 # B1c B2c w0 Ac 0 Ac 0 y ¼ ½ C2c C2r sI þ ½ C2c C2r sI B1r B2r u 0 Ar 0 Ar ð20Þ Comparing Eq. (20) with Eq. (18), one obtains the unmodelled dynamics between the full model and the nominal model in the frequency domain as
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DP21 ¼ C2r ðsI Ar Þ1 B1r DP22 ¼ C2r ðsI Ar Þ1 B2r and the term DP22 is the main reason to cause control spillover. Introduce the suppositional evaluation output zW ¼ W u and the suppositional input disturbance wW ¼ D zW where W is the weighting function, D is a coordinate matrix which satisfy the follow conditions D W ¼ DP22 ;
kDk1 6 1
From Ref. [20] one knows that the robust stability condition of the closed loop system can be expressed as kTzW wW k1 ¼ kW Hk1 < 1
ð21Þ
where H is the closed loop transfer function from the output of DP22 to the control input u which can be expressed as H ¼ ðI GP22 Þ1 G 4.3. H1 controller design If the state space form expression of W is (AW , BW , CW , DW ), then the augmented controlled system can be expressed as x_ ¼ Ax þ B1 w þ B2 u
ð22Þ
z ¼ C1 x þ D12 u
ð23Þ
y ¼ C2 x þ D21 w
ð24Þ
where
w0 z0 Ac xc ; w¼ ; z¼ ; A¼ xW wW zW 0 B1c 0 B2c C1c 0 ; B2 ¼ B1 ¼ ; C1 ¼ ; 0 0 0 CW BW D12c ; C2 ¼ ½ C2c 0 ; D21 ¼ ½ 0 I D12 ¼ DW
x¼
0 ; AW
Solve the above standard H1 problem, one can obtain the state space form output feedback robust controller G x_ G ¼ AG xG þ B1 w þ BG y u ¼ CG xG
ð25Þ
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where AG ¼ A þ B1 BT1 X Z1 LC2 þ B2 F BG ¼ Z1 L;
CG ¼ E1 F
F ¼ BT2 X;
L ¼ YCT2 ;
Z ¼ I XY
where positive definite matrix E satisfies relation DT12 D12 ¼ EE. Matrices X and Y are the positive semi-definite solutions of the following two Riccati equations
1 T T T B1 B1 B2 B2 þ CT1 C1 ¼ 0 ð26Þ A X þ XA þ X e
1 AY þ YAT þ Y CT1 C1 CT2 C2 Y þ B1 BT1 ¼ 0 e The closed loop equation of the system can be expressed as 8 9 2 38 9 2 3 0 A B2 CG B1 < x_ = < x = AG BG C2r 5 xG þ 4 B1 þ BG D21 5w x_ ¼ 4 BG C2c : G; : ; 0 B2r CG Ar xr ½ B1r 0 x_ r
ð27Þ
ð28Þ
If the unmodelled dynamics is not considered, the controller will be the classical H1 controller.
5. Numerical simulation research In order to compare study these three proposed control methodology, a computer simulative research is carried out on a four-bar linkage mechanism shown in Fig. 1. The property parameters of the mechanism and the PZT piezoelectric ceramic actuators and sensors are shown in Tables 1 and 2. A pair of actuator and a pair of sensor are bonded on each of the links. The actuators and sensors on the crank link are 30 mm long and 30 mm wide and are located at a quarter and three quarters length of the link while those on the coupler and the follower 40 mm long and 25 mm
Fig. 1. Four-bar linkage.
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Table 1 Parameters of the mechanism links Parameters
Link 1
Link 2
Link 3
Link 4
Length (mm) Width (mm) Thickness (mm)
98.8 30 3
302.0 25 2
300.0 25 2
332.2 – –
Table 2 The material properties Properties
Links
Actuators
Sensors
Density kg/m3 Young’s modulus (N/m2 ) Poisson’s ratio Dielectric constant 8:85 1012 F/m Piezoelectric constant m/V
2700 7 1010 0.25 – –
7500 1:17 1011 0.25 1700 185:0 1012
1760 0:15 1010 0.25 12 20:0 1012
wide, and at their three eighths and five eighths length. In this study, all the links are treated as flexible. Frame element is used in the mechanism. The crank is modeled by two finite elements, and the coupler and follower are both modeled by four elements, so the system has 30° of freedom. The first five modes are retained to calculate the response of the system while the first three modes are taken as controlled modes. The crank speed is 398 rpm. The lower order resonance [21] will be occurred at this speed for this system. The damping ratios of the controlled modes are set at 0.03. The allowable voltage of the piezoelectric actuators used in the mechanism system is from 500 to þ500 V. To meet the voltage limit, properly selecting the coefficient b for the reduced modal controller design, and the weighting matrices Q and R, as well as the weighting function W for the H1 controller design are very important. After a certain amount of trail and error, we select the coefficient b as 0.7, and select the weighting matrices Q and R as Q ¼ diag½ ; 108 ; ;
R ¼ diag½2:0; 1:2; 2:0 ; and e ¼ 6:4 109
and the weighting functions as s
8:0 1013 900 þ1 W ¼
2 s s þ 2 0:02 1500 þ1 1500 From Section 2, we have the relation x x ¼ ½ Xc Xr c xr where Xc and Xr are the eigenvector matrices corresponding to the controlled and residual modes. Considering Eqs. (7), (14), and (28), one can obtain the transfer function Txw0 of following cases: for the original system 1 Ac 0 B1c Txw0 ¼ ½ Xc Xr sI B1r 0 Ar
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for the reduced modal control system
Txw0 ¼ ½ Xc
Xr sI
Ac þ DAc
B2c F1 Cr
B2r F1 Cc
Ar þ B2r F1 Cr
0
A
1
ð1 bÞB1c
B1r þ B2r F2
for the H1 control system
Txw0 ¼ ½ ½Xc
0
0
2
B 6 Xr @sI 4 BG C2c 0
B2 CG AG B2r CG
0
311 2
½B1c
0
½B1r
0
3
7 7C 6 BG C2r 5A 4 B1 þ BG D21 5 Ar
Fig. 2 shows the frequency response of the open loop and the closed loop system. The input is frequency and the output is the maximum singular values of the transfer functions in one cycle of operation for the mechanism. From this figure, one finds that the frequency response has been entirely changed by the H1 controller. Control spillover is occurred at the location of the fourth and the fifth modes for the reduced mode control method and the standard H1 method, and the spillover is avoided for the robust H1 method. Fig. 3 shows the transient elastodynamic response of the midpoint of the coupler link both with and without control. It is seen that the significant vibration suppression is achieved after control. The response settles to steady state much faster and the steady-state responses are remarkably suppressed after employing the reduced mode control method. This indicates that the reduced mode controller increase modal damping efficiently for the system. The H1 method cannot damp out the transient vibration as efficient as that of the reduced mode method, but the method can control the steady-state response efficiently. From Fig. 4, one finds that at the same level of control for vibration, the input voltage is higher and changes more violent than that of the robust H1 method.
Fig. 2. Frequency response. (––) Original; ( ) reduced mode; (– – –) standard H1 ; (
) robust H1 .
Z. Xianmin et al. / Mechanism and Machine Theory 37 (2002) 985–997
Fig. 3. Transient response of the coupler in x-direction (a) and y-direction (b). ( ) Original system; ( H1 control; (––) reduced mode control.
995
) robust
Fig. 4. Input voltages of the actuator on the coupler. (––) Robust H1 control; ( ) reduced mode control.
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6. Conclusions The reduced modal controller, the classical and the robust H1 controller for active vibration control of the high-speed flexible linkage mechanism systems with piezoelectric actuators and sensors are investigated. Simulation researches show that the vibration of the system is significantly suppressed with permitted actuator voltages by each of these controllers. The reduced modal control makes it more convenient to increase the modal damping of the system than that of the H1 control does. This method is especially fit for control the transient vibrations of the system. The robust H1 control can avoid the spillover due to mode truncation and is especially fit for control the steady-state response of the mechanism system. At the same level of control for vibration, the input voltage is higher and changes more violent than that of the robust H1 method. Acknowledgements This research was supported by the National Natural Science Foundation of China under grants 59975056 and 59605001, the Foundation for University Key Teacher by the Ministry of Education of China, the Natural Science Foundation of Guangdong Province under grant 20000783 and 970381, and the elitist Foundation of Guangdong Province under grant 9912. The first author gratefully acknowledges these support agencies.
References [1] X.M. Zhang, Y.W. Shen, H.Z. Liu, W.Q. Cao, Optimal design of flexible mechanisms with frequency constraints, Mechanism and Machine Theory 30 (1) (1995) 131–139. [2] W.L. Cleghorn, F.G. Fenton, B. Tabarrok, Optimal design of high-speed flexible mechanisms, Mechanism and Machine Theory 16 (1981) 339–406. [3] E.H. El-Dannah, S.H. Farghaly, Vibratory response of a sandwich link in a high-speed mechanism, Mechanism and Machine Theory 28 (1993) 447–457. [4] X.M. Zhang, Modal loss factor predication in the passive vibration control of elastic mechanism systems, Chinese Journal of Mechanical Engineering 35 (3) (1999) 55–58. [5] Carl Sisemore, Ahmad Smaili, Richard Houghton, Passive damping of flexible mechanism systems: experimental and finite element investigations, vol. 5, The 10th World Congress on the Theory of Machines and Mechanisms, Oulu, Finland, 20–24 June 1999, pp. 2140–2145. [6] A. Ghazavi, F. Gardanine, N.G. Chalhout, Dynamic analysis of a composite material flexible robot arm, Computers and Structures 49 (1993) 315–325. [7] C.K. Sung, B.S. Thompson, Material selection: an important parameter in the design of high-speed linkages, Mechanism and Machine Theory 19 (1984) 389–396. [8] J.H. Oliver, D.A. Wysocki, B.S. Thompson, The synthesis of flexible linkages by balancing the tracer point quasistatic deflections using microprocessor and advanced material technologies, Mechanism and Machine Theory 20 (1985) 471–482. [9] K. Soong, D. Sunappan, B.S. Thompson, The elastodynamic response of a class of intelligent machinery, part 1: theory, ASME Journal Vibration Acoustics, and Reliability in Design 111 (1989) 430–436. [10] K. Soong, D. Sunappan, B.S. Thompson, The elastodynamic response of a class of intelligent machinery, part 2: computional and experimental results, ASME Journal Vibration Acoustics, and Reliability in Design 111 (1989) 437–442.
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[11] C.K. Sung, Y.C. Chen, Vibration control of the elastodynamic response of high-speed flexible linkage mechanisms, ASME Journal of Vibration and Acoustics 113 (1991) 14–21. [12] C.Y. Liao, C.K. Sung, An elastodynamic analysis and control of flexible linkages using piezoceramic sensors and actuators, ASME Journal of Mechanical Design 115 (1993) 658–665. [13] S.B. Choi, C.C. Cheong, B.S. Thompson, M.V. Gandhi, Vibration control of flexible linkage mechanisms using piezoelectric films, Mechanism and Machine Theory 29 (1994) 535–546. [14] X.M. Zhang, Active residual vibration control for flexible linkage mechanisms, Journal of Vibration Engineering 12 (1999) 93–99. [15] X.M. Zhang, H.Z. Liu, W.Q. Cao, Active vibration control of flexible mechanisms, Chinese Journal of Mechanical Engineering 32 (1996) 7–14. [16] B.S. Thompson, X. Tao, A note on the experimentally determined elastodynamic response of a slider crank mechanism featuring a macroscopically smart connecting rod with ceramic piezoelectric actuators and strain gauge sensors, 1994 Machine Element and Machine Dynamics De-71, The 1994 ASME Design Technique Conferences, Minneapolis, Minnesota, 11–14 September 1994, pp. 63–69. [17] M. Sannah, A. Smaili, Active control of elastodynamic vibrations of a four-bar mechanism system with a smart coupler link using optimal multivariable control: experimental implementation, ASME Journal of Mechanical Design 120 (1998) 316–326. [18] C.J. Shao, X.M. Zhang, Y.W. Shen, Complex mode active vibration control of high-speed flexible linkage mechanisms, Journal of Sound and Vibration 234 (2000) 491–506. [19] X.M. Zhang, C.J. Shao, Active Vibration Control for the Closed Loop Flexible Mechanisms Combined the Feedback and Feed-Forward Strategy, Part 1: Modelling, 2000 ASME Design Engineering Technical Conferences, Baltimore, Maryland, DETC 2000/MECH-14075. [20] B.A. Francis, A Course in H1 Control Theory, Springer-Verlag, New York, 1987. [21] X.M Zhang, J.K. Liu, Y.W. Shen, Lower order critical operating speeds identification for flexible linkage mechanisms, Journal of Vibration Engineering 9 (1996) 99–104.
Active vibration controller design and comparison study of flexible linkage mechanism systems Zhang Xianmin
a,*
, Shao Changjian a, Arthur G. Erdman
b
a
b
Department Mechatronic Engineering, Shantou University, Shantou, Guangdong 515063, PR China Department of Mechanical Engineering, University of Minnesota, 111 Church Street S.E., Minneapolis, MN 55455-0111, USA Received 23 March 2001; accepted 11 February 2002
Abstract Three active vibration controller design methodologies include the reduced modal controller, the classical and the robust H1 controller for the high-speed flexible linkage mechanism systems with piezoelectric actuators and sensors are investigated. Firstly, the state space form control model is formulated based on the complex mode theory. Secondly, the reduced modal controller, the classical and the robust H1 controller are designed. Finally, numerical simulations are carried out on a 4-bar linkage mechanism. The simulation results show that the vibration of the system is significantly suppressed with permitted actuator input voltages by each of the three controllers. The reduced modal control makes it more convenient to increase the modal damping of the system than that of the H1 control does, while the robust H1 control can avoid the spillover due to mode truncation. Ó 2002 Published by Elsevier Science Ltd. Keywords: Active vibration control; Flexible linkage mechanism; Modal; Controller; Actuator; Sensor
1. Introduction Recently, considerable attention has been paid to the investigation of the vibration control for high-speed flexible mechanism systems. Heretofore, there are basically five design philosophies, which have been developed to improve the elastodynamic responses for these kinds of systems. The first involves designing links of conventional materials with optimization of the cross-sectional geometry of the members [1,2]. The second one uses additional damping materials to
*
Corresponding author. Tel./fax: +86-754-2903313. E-mail address: [email protected] (Z. Xianmin).
0094-114X/02/$ - see front matter Ó 2002 Published by Elsevier Science Ltd. PII: S 0 0 9 4 - 1 1 4 X ( 0 2 ) 0 0 0 2 5 - 3
986
Z. Xianmin et al. / Mechanism and Machine Theory 37 (2002) 985–997
Nomenclature system damping matrix system control matrix and output matrices system generalized force vector coefficient matrices output feedback robust controller transfer function identity matrix system stiffness matrix system mass matrix the number of the degrees of freedom of the system the number of the actuators and sensors bonded on an element the number of controlled and uncontrolled modes weighting matrices nodal displacement, velocity, and acceleration vectors of the whole linkage in the global frame u control input vector W weighting function suppositional input disturbance wW X, X0 the left and right eigenvector matrices x state space vector modal coordinate vector x0 D coordinate matrix b counteract coefficient e a given constant kiR , kiI par the real and image part of the ith complex eigenvalue C Da , Ds F F1 , F2 G H I K M n na , ns nc , nr Q, R q, q_ , € q
dissipate the vibration energy [3–5]. The third advocates that the mechanism links should be built of advanced composite materials because of their high damping and high stiffness-to-weight ratio [6,7]. These three design concepts may be referred to as passive vibration control. The fourth introduces a microprocessor-controlled actuator into the original mechanism to reduce the deflection of the flexible linkage [8–10]. The last involves the application of smart materials featuring distributed actuators and sensors to the linkage mechanisms to control unwanted vibration [11– 18]. The advantage of this method is that the controller can adapt with system changes, and thus can be much more effective than other methods. Sung and Chen [11] attempted to control the elastodynamic response of a four-bar linkage mechanism system with a flexible follower, a rigid crank, and a rigid coupler. Two patches of piezoceramic actuators were bonded to the surface of the follower at two locations that are symmetric with respect to its midpoint and one piezoceramic sensor was bonded to the surface at its midpoint. The authors assumed that the free vibration modes of the flexible follower link were the same as those of a simply supported beam. Optimal linear quadratic regulator (LQR) is employed in the control system. Liao and Sung [12] derived
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the finite element equations of the above mechanism and studied the active vibration control problem both analytically and experimentally based on the linear quadratic Gaussian theory. In the work of Ref. [13], Choi et al. investigated a slider-crank mechanism whose flexible connecting rod was bonded with two piezoelectric films to its upper and lower surface. One of the films was used as a distributed sensor while the other was used as a distributed actuator. The linear optimal feedback control law with a Luenberger observer was employed. On the basis of the independent modal control method, Zhang et al. [14,15] studied the active vibration control problem for the flexible mechanisms all of whose members were considered as flexible. By means of experimental studies, Thompson and Tao [16] controlled the connecting rod vibrations of a slider crank mechanism by using two sets of piezoelectric actuator/strain gauge sensor pairs. Classical position feedback controller was employed in the experimental system. Sannah and Smaili [17] presented an experimental investigation for controlling the elastodynamic response of a four-bar mechanism system with a flexible coupler link, slightly less flexible follower link, and relatively rigid crank. A controller, which consists of a LQR and a Luenberger observer, was designed and implemented. However, the state space matrices of the system were assumed to be constant for the entire motion cycle of the mechanism, which is not the case when the mechanism operates at high speeds. In Ref. [18], Shao and Zhang develop a hybrid independent modal controller which is composed of state feedback and disturbance feed-forward control laws. Like other methods, the control spillover problem may exist if the control is not imputed properly. In this paper, based on the complex mode theory and the control model developed in Ref. [19], the reduced modal controller, the classical H1 , and the robust H1 controller for active vibration control of the high-speed flexible linkage mechanism systems with piezoelectric actuators and sensors are investigated. Comparison study shows that the vibration of the system is significantly suppressed with permitted actuator voltages by each of these controllers. The reduced modal control makes it more convenient to increase the modal damping of the system than that of the H1 control does, while the robust H1 control avoids the spillover due to mode truncation. At the same level of control for vibration, the input voltage is higher and changes more violent than that of the robust H1 method. 2. Modelling of control According to the Hamilton theory, the mathematical model of control for the flexible linkage mechanism can be expressed as [19] M€ q þ Cq_ þ Kq ¼ F Da u
ð1Þ
y ¼ Ds q
ð2Þ
where M, C, K, Da , and Ds are the systematic mass, damping, stiffness, control and output matrices, respectively; F is the systematic generalized force vector; q, q_ , €q are the generalized displacement, the velocity, and acceleration vectors of the linkage system, respectively; u is the control input vector; and y is the output vector of the sensors. This model includes both the rigid body and the elastic motion coupling terms and the elastodynamics and piezoelectricity coupling terms, and takes into account the effects of the piezoelectric apparatus upon the mass and the stiffness of the system. It is noted that all the system matrices are not constant but periodically
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varying functions of the position of each links of the mechanism. Matrices C and K are not symmetric matrices, so the equations cannot be solved using the traditional real modal method. In order to decouple Eq. (1), we define a state space vector as x ¼ fq_ T ; qT gT ;
x_ ¼ f€ qT ; q_ T gT
Rewriting Eqs. (1) and (2) in terms of the state space vector yields Ax_ þ Bx ¼ w0 þ Du
ð3Þ
y ¼ Sx
ð4Þ
where
0 A¼ M
M ; C
M 0 B¼ ; 0 K
0 0 ; w0 ¼ D¼ F Da
S ¼ ½0
Ds
Based on the complex modal theory, one can obtain the state space form de-coupled equations of motion as [18] x_ 0 ¼ A0 x0 þ B10 w0 þ B20 u
ð5Þ
y ¼ C20 x0
ð6Þ
where
A0 ¼ Block diagð Aii Þ; C20 ¼ SX;
Aii ¼
kiR
kiI
kiI
kiR
B10 ¼ X0T ;
B20 ¼ X0T D;
i ¼ 1; 2; . . . ; n
where kiR and kiI i ¼ 1; 2; . . . ; n are the real and image part of the complex eigenvalues, n is the number of the generalized coordinates of the system, terms X and X0 indicate the normalized left and the right eigenvector matrices, respectively. From the control theory, one can consider the first number of nc vibration modes as controlled modes and leave a number of nr modes as uncontrolled modes with nc þ nr ¼ n. Let x0 ¼ fxTc
xTr gT
Eqs. (5) and (6) can be rewritten as B2c Ac 0 xc B1c x_ c ¼ þ w þ u xr B1r 0 B2r 0 Ar x_ r x y ¼ ½ C2c C2r c xr
ð7Þ ð8Þ
3. The reduced modal controller design The reduced modal control indicates that the controller is designed based on the controlled modes. Dividing the control input into two parts u ¼ u1 þ u2
ð9Þ
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where u1 is the state feedback term which allow to assignment the poles of the system to provide sufficient damping for the controlled modes, and u2 is the feed-forward terms which is used to counteract the disturbance input of the system. The two terms can be expressed as u1 ¼ F1 y;
u2 ¼ F2 w0
ð10Þ
where F1 , F2 are the state feedback and the disturbance feed-forward coefficient matrices, when we just consider the control modes, they satisfy following relations B2c F1 C2c ¼ DAc ;
B2c F2 ¼ bB1c
ð11Þ
where matrix DAc consists of the difference between the original and the required eigenvalues of the controlled modes, b is the counteract coefficient of the input disturbance. Properly selecting the positions of the actuators and the sensors, the matrices B2c and C2c can be full row ranked, then the first nc th modes are controllable and observable. Let the numbers of the actuators and the sensors be equal to nc . Solving the above equations one can obtain the state feedback and disturbance feed-forward coefficient matrices F1 , F2 as F1 ¼ ðBT2c B2c ÞT BT2c DAc CT2c ðC2c CT2c Þ1
ð12Þ
F2 ¼ bðBT2c B2c Þ1 BT2c B1c
ð13Þ
Substituting the two terms into Eqs. (7) and (8), considering the residual modes, one can further obtain the closed-loop equation as B2c F1 C2r Ac þ DAc xc ð1 bÞB1c x_ c ð14Þ ¼ þ w B2r F1 C2c Ar þ B2r F1 C2r xr B1r þ B2r F2 0 x_ r
4. Classical and robust H‘ controller design 4.1. The design performance index and the nominal control model The aim of this section is to design a controller u ¼ Gy
ð15Þ
which satisfies the following conditions (1) the closed-loop system is stable; (2) when the initial state vector xð0Þ ¼ 0, then at any time t the system satisfy the relation. Z t Z t T T J¼ ðx Qx þ u RuÞ dt < e wT0 w0 dt ð16Þ 0
0
where e is a given constant, Q and R are the weighting matrices. Definite the controlled output vector as z0 ¼ Nx þ Lu where Q ¼ NT N, R ¼ LT L.
ð17Þ
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When we just consider the controlled modes, the output vector can be expressed as z0 ¼ C1c xc þ D12c u where C1c ¼ NXc , D12c ¼ L, and Xc is the eigenvector matrix corresponding to the controlled modes. The equivalent form of Eq. (16) is Z t zT0 z0 dt J¼ 0
If we just consider the controlled modes in the performance index, then the nominal transfer function from the input vector to the output vector is
and the transfer function form of Eqs. (7) and (8) can be expressed as z0 P11 P12 w0 w0 ¼ ¼P y0 u P21 P22 u
ð18Þ
where P11 ¼ C1c ðsI Ac Þ1 B1c ; 1
P21 ¼ C2c ðsI Ac Þ B1c ;
P12 ¼ C1c ðsI Ac Þ1 B2c þ D12c 1
P22 ¼ C2c ðsI Ac Þ B2c
Then the equivalent form of Eq. (16) can be further expressed as pffiffi Tz w < c ¼ e 0 0 1
ð19Þ
where Tz0 w0 is the closed loop transfer function from w0 to z0 , k k1 indicates the H1 norm. Tz0 w0 ¼ P11 þ P12 GðI P22 GÞ1 P21 4.2. The unmodelled dynamics and the robust stability index The transfer function form of the full model (7–8) can be expressed as " 1 1 # B1c B2c w0 Ac 0 Ac 0 y ¼ ½ C2c C2r sI þ ½ C2c C2r sI B1r B2r u 0 Ar 0 Ar ð20Þ Comparing Eq. (20) with Eq. (18), one obtains the unmodelled dynamics between the full model and the nominal model in the frequency domain as
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DP21 ¼ C2r ðsI Ar Þ1 B1r DP22 ¼ C2r ðsI Ar Þ1 B2r and the term DP22 is the main reason to cause control spillover. Introduce the suppositional evaluation output zW ¼ W u and the suppositional input disturbance wW ¼ D zW where W is the weighting function, D is a coordinate matrix which satisfy the follow conditions D W ¼ DP22 ;
kDk1 6 1
From Ref. [20] one knows that the robust stability condition of the closed loop system can be expressed as kTzW wW k1 ¼ kW Hk1 < 1
ð21Þ
where H is the closed loop transfer function from the output of DP22 to the control input u which can be expressed as H ¼ ðI GP22 Þ1 G 4.3. H1 controller design If the state space form expression of W is (AW , BW , CW , DW ), then the augmented controlled system can be expressed as x_ ¼ Ax þ B1 w þ B2 u
ð22Þ
z ¼ C1 x þ D12 u
ð23Þ
y ¼ C2 x þ D21 w
ð24Þ
where
w0 z0 Ac xc ; w¼ ; z¼ ; A¼ xW wW zW 0 B1c 0 B2c C1c 0 ; B2 ¼ B1 ¼ ; C1 ¼ ; 0 0 0 CW BW D12c ; C2 ¼ ½ C2c 0 ; D21 ¼ ½ 0 I D12 ¼ DW
x¼
0 ; AW
Solve the above standard H1 problem, one can obtain the state space form output feedback robust controller G x_ G ¼ AG xG þ B1 w þ BG y u ¼ CG xG
ð25Þ
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where AG ¼ A þ B1 BT1 X Z1 LC2 þ B2 F BG ¼ Z1 L;
CG ¼ E1 F
F ¼ BT2 X;
L ¼ YCT2 ;
Z ¼ I XY
where positive definite matrix E satisfies relation DT12 D12 ¼ EE. Matrices X and Y are the positive semi-definite solutions of the following two Riccati equations
1 T T T B1 B1 B2 B2 þ CT1 C1 ¼ 0 ð26Þ A X þ XA þ X e
1 AY þ YAT þ Y CT1 C1 CT2 C2 Y þ B1 BT1 ¼ 0 e The closed loop equation of the system can be expressed as 8 9 2 38 9 2 3 0 A B2 CG B1 < x_ = < x = AG BG C2r 5 xG þ 4 B1 þ BG D21 5w x_ ¼ 4 BG C2c : G; : ; 0 B2r CG Ar xr ½ B1r 0 x_ r
ð27Þ
ð28Þ
If the unmodelled dynamics is not considered, the controller will be the classical H1 controller.
5. Numerical simulation research In order to compare study these three proposed control methodology, a computer simulative research is carried out on a four-bar linkage mechanism shown in Fig. 1. The property parameters of the mechanism and the PZT piezoelectric ceramic actuators and sensors are shown in Tables 1 and 2. A pair of actuator and a pair of sensor are bonded on each of the links. The actuators and sensors on the crank link are 30 mm long and 30 mm wide and are located at a quarter and three quarters length of the link while those on the coupler and the follower 40 mm long and 25 mm
Fig. 1. Four-bar linkage.
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Table 1 Parameters of the mechanism links Parameters
Link 1
Link 2
Link 3
Link 4
Length (mm) Width (mm) Thickness (mm)
98.8 30 3
302.0 25 2
300.0 25 2
332.2 – –
Table 2 The material properties Properties
Links
Actuators
Sensors
Density kg/m3 Young’s modulus (N/m2 ) Poisson’s ratio Dielectric constant 8:85 1012 F/m Piezoelectric constant m/V
2700 7 1010 0.25 – –
7500 1:17 1011 0.25 1700 185:0 1012
1760 0:15 1010 0.25 12 20:0 1012
wide, and at their three eighths and five eighths length. In this study, all the links are treated as flexible. Frame element is used in the mechanism. The crank is modeled by two finite elements, and the coupler and follower are both modeled by four elements, so the system has 30° of freedom. The first five modes are retained to calculate the response of the system while the first three modes are taken as controlled modes. The crank speed is 398 rpm. The lower order resonance [21] will be occurred at this speed for this system. The damping ratios of the controlled modes are set at 0.03. The allowable voltage of the piezoelectric actuators used in the mechanism system is from 500 to þ500 V. To meet the voltage limit, properly selecting the coefficient b for the reduced modal controller design, and the weighting matrices Q and R, as well as the weighting function W for the H1 controller design are very important. After a certain amount of trail and error, we select the coefficient b as 0.7, and select the weighting matrices Q and R as Q ¼ diag½ ; 108 ; ;
R ¼ diag½2:0; 1:2; 2:0 ; and e ¼ 6:4 109
and the weighting functions as s
8:0 1013 900 þ1 W ¼
2 s s þ 2 0:02 1500 þ1 1500 From Section 2, we have the relation x x ¼ ½ Xc Xr c xr where Xc and Xr are the eigenvector matrices corresponding to the controlled and residual modes. Considering Eqs. (7), (14), and (28), one can obtain the transfer function Txw0 of following cases: for the original system 1 Ac 0 B1c Txw0 ¼ ½ Xc Xr sI B1r 0 Ar
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for the reduced modal control system
Txw0 ¼ ½ Xc
Xr sI
Ac þ DAc
B2c F1 Cr
B2r F1 Cc
Ar þ B2r F1 Cr
0
A
1
ð1 bÞB1c
B1r þ B2r F2
for the H1 control system
Txw0 ¼ ½ ½Xc
0
0
2
B 6 Xr @sI 4 BG C2c 0
B2 CG AG B2r CG
0
311 2
½B1c
0
½B1r
0
3
7 7C 6 BG C2r 5A 4 B1 þ BG D21 5 Ar
Fig. 2 shows the frequency response of the open loop and the closed loop system. The input is frequency and the output is the maximum singular values of the transfer functions in one cycle of operation for the mechanism. From this figure, one finds that the frequency response has been entirely changed by the H1 controller. Control spillover is occurred at the location of the fourth and the fifth modes for the reduced mode control method and the standard H1 method, and the spillover is avoided for the robust H1 method. Fig. 3 shows the transient elastodynamic response of the midpoint of the coupler link both with and without control. It is seen that the significant vibration suppression is achieved after control. The response settles to steady state much faster and the steady-state responses are remarkably suppressed after employing the reduced mode control method. This indicates that the reduced mode controller increase modal damping efficiently for the system. The H1 method cannot damp out the transient vibration as efficient as that of the reduced mode method, but the method can control the steady-state response efficiently. From Fig. 4, one finds that at the same level of control for vibration, the input voltage is higher and changes more violent than that of the robust H1 method.
Fig. 2. Frequency response. (––) Original; ( ) reduced mode; (– – –) standard H1 ; (
) robust H1 .
Z. Xianmin et al. / Mechanism and Machine Theory 37 (2002) 985–997
Fig. 3. Transient response of the coupler in x-direction (a) and y-direction (b). ( ) Original system; ( H1 control; (––) reduced mode control.
995
) robust
Fig. 4. Input voltages of the actuator on the coupler. (––) Robust H1 control; ( ) reduced mode control.
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6. Conclusions The reduced modal controller, the classical and the robust H1 controller for active vibration control of the high-speed flexible linkage mechanism systems with piezoelectric actuators and sensors are investigated. Simulation researches show that the vibration of the system is significantly suppressed with permitted actuator voltages by each of these controllers. The reduced modal control makes it more convenient to increase the modal damping of the system than that of the H1 control does. This method is especially fit for control the transient vibrations of the system. The robust H1 control can avoid the spillover due to mode truncation and is especially fit for control the steady-state response of the mechanism system. At the same level of control for vibration, the input voltage is higher and changes more violent than that of the robust H1 method. Acknowledgements This research was supported by the National Natural Science Foundation of China under grants 59975056 and 59605001, the Foundation for University Key Teacher by the Ministry of Education of China, the Natural Science Foundation of Guangdong Province under grant 20000783 and 970381, and the elitist Foundation of Guangdong Province under grant 9912. The first author gratefully acknowledges these support agencies.
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