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Active vibration control of a flexible one-link manipulator using a multivariable predictive controller
Mechatronics 17 (2007) 311–323
Active vibration control of a flexible one-link manipulator using a multivariable predictive controller M. Hassan *, R. Dubay, C. Li, R. Wang Mechanical Engineering Department, University of New Brunswick, Fredericton, Canada Received 13 October 2005; accepted 26 February 2007
Abstract This article presents a new application of model-based predictive controller (MPC) for vibration suppression of a flexible one-link manipulator using piezoceramic actuators. Simulation and experimental studies were conducted to investigate the applicability of the MPC strategy to control vibration of the flexible structure having multiple inputs and multiple outputs (MIMO). The performance of the proposed technique was assessed in terms of level of vibration reduction. The results demonstrated that the proposed predictive control strategy is well suited for multivariable control of vibration suppression on flexible structures. 2007 Elsevier Ltd. All rights reserved. Keywords: Active vibrations control; Mechatronics; Smart structures
1. Introduction The interdisciplinary area of active–passive damping of flexible structures has been covered by a large number of excellent research papers due to its high potential for industrial applications. Over the past two decades, piezoelectric materials for sensing and control of vibration on flexible structures have been well-studied [1], with [2] the first to suggest this idea. For small amplitude vibration on very flexible structures, active approaches lead to lightweight and high performance control systems [1]. Conversely, pure passive techniques using layers of viscoelastic materials are low cost, reliable and can provide good vibration control [3]. This approach is widely used in aerospace, automobile, and aeronautical industries. Several investigations [4–6] have focused on the simultaneous use of piezoelectric (active) and viscoelastic (passive) materials for adaptive and robust vibration control. These methods can employ simultaneous or separate active and
*
Corresponding author. Tel.: +1 506 458 7248; fax: +1 506 453 5025. E-mail address: [email protected] (M. Hassan).
0957-4158/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechatronics.2007.02.004
passive actions. Shen [7,8] used active piezoelectric damping and optical sensors to measure vibration displacements of a beam. Varadan et al. [9] introduced the approach of a piezoelectric sensor bonded next to the actuator. Baz [10] introduced the concept of using two symmetrical bonded sets of segmented piezoelectric sensors and actuators on a flexible beam. Bailey and Hubbard [11] used distributed-parameter control theory and a piezoceramic actuator to actively control vibration on a cantilever beam actively. A sliding mode state feedback controller developed by [12] used a piezoelectric actuator for force tracking of a two-fingered flexible gripper. Other researchers [13,14] studied the effect of the actuators on the host structures for vibration control through modal shape analysis. A variable structure adaptive controller developed by [15] to control contact forces on a cantilever beam used only the output force as feedback, resulting in undesirable chattering. Artificial neural networks (ANN) for identification and state feedback control of flexible structures have been implemented with good preliminary results [16]. Robust control focuses on the ability to have good control performance and stability in the presence of uncertainty in the system model as well as its
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M. Hassan et al. / Mechatronics 17 (2007) 311–323
Nomenclature a d f h k q / x Du Nnc WnVi ai d31 e fm fps fPin kh ku mii ns nu q ~q r s tb tc u v vm vnp w x y y0 ysp b y
tunable constant in PDT adjustment parameter modal damping coefficient joint angle tuning parameter material density mode shapes natural frequency control move vector increment shape function of the nth PVDF film ith modal coefficient of PVDF step (or impulse) response value transverse piezoelectric charge to stress ratio error modal force coefficient for motor modal force coefficient for structure and PZT modal force coefficient of PZT potentiometer encoder gain motor torque constant elements of mass matrix number of PVDF sensors control horizon modal coordinates generalized coordinates axial location projected desired trajectory (PDT) of motor position thickness of the beam thickness of the piezoelectric film control move vector beam velocity control voltage applied to the motor control voltage applied to the nth PZT actuator flexural beam deflection state vector system output vector initial condition of the measured output setpoint trajectory predicted process profile
exogenous inputs, including disturbances and noise. The H1 controller compensates for some of these uncertainties in active vibration control [17]. Recently, [18] developed a robust rejection method using a Kalman filter to estimate the system states under persistent excitation. The investigations in control all lack predictability of the flexible structures’ vibrations and dynamic response to multi-actuators or control actions, disturbances and noise. This prediction can be useful for providing tight and robust control performance, as the control actions can be evaluated using these predictions. A large amount of research has focused on the optimization of sensors/actuators numbers and location, an example being [19]. The use of a pre-
zc zi A Acs As Bm Bs Cc Cs D E Eb Ec F nPi Hs Hps I Ics J Jh Ks Kps K nP L Ms Mps N NPZT P Q Qs R V noc Vh Wb Wc
distance from the neutral axis of the beam to the piezoelectric film backward shift operator in z-transform dynamic matrix cross-sectional area of the beam state matrix motor viscous friction coefficient input matrix capacitance of the piezoelectric film output matrix disturbance vector of future errors Young’s modulus of the beam Young’s modulus of the PZT film generalized nth PZT force for the ith mode modal damping matrix for structure modal damping matrix for structure and PZT unity matrix moment of inertia of the beam controller objective function (or performance index) motor-fixture inertia modal stiffness matrix for structure modal stiffness matrix for structure and PZT stiffness matrix of the nth PZT film length of the beam modal mass matrix for structure modal mass matrix for structure and PZT number of modes number of PZT actuators controller predictive horizon weighting matrix for error vector generalized force vector weighting matrix for control move vector open circuit voltage of nth PVDF patch potentiometer voltage signal width of the beam width of the piezoelectric film
dictive control (PC) strategy can address the optimization of sensors/actuators numbers and location due to its robust and predictive architecture, without the need for extensive computational optimization algorithms. Predictive control has been used on very difficult to control systems that are highly nonlinear having time variant properties [26]. Wang et al. [27] have very recently introduced PC for vibration suppression on a motor driven flexible beam, with very good simulation and practical results. These initial results strongly demonstrate that PC can effectively provide good active suppression vibration on a flexible beam. To the best of the authors knowledge, the use of PC for active vibration suppression is almost non-existent, making it a prime
M. Hassan et al. / Mechatronics 17 (2007) 311–323
area of research to explore, develop and implement on practical systems.
Y
Z θ (t) r, t)
2. Modeling of flexible beam dynamics
wðr; tÞ ¼
/i ðrÞqi ðtÞ
ð1Þ
i¼1
where N is the number of modes. The velocity at any location r on the beam is given by _ _ _ tÞ ¼ hðtÞr vðr; tÞ ¼ hðtÞr þ wðr; þ
N X
/i ðrÞq_ i ðtÞ
ð2Þ
i¼1
where denotes differentiation with respect to t, h is joint angle and rh is the rigid-body displacements of the beam.
2
3
J h þ qAcs3 L 6 R 6 qAcs L / ðrÞr dr 1 6 0 6 RL 6 M s ¼ 6 qAcs /2 ðrÞr dr 0 6 6 .. 6 4 . RL qAcs 0 /N ðrÞr dr
qAcs qAcs
RL 0
RL 0
/1 ðrÞr dr
qAcs
RL 0
/21 ðrÞ dr 0 .. .
qAcs
RL 0
0 60 6 6 6 Ks ¼ 6 0 6. 6. 4. 0
0 RL Eb I cs 0 ð/001 ðrÞÞ2 dr 0 .. . 0
Eb I cs
RL 0
0 0 ð/002 ðrÞÞ2 dr .. . 0
Fig. 1. Schematic of the system.
The torque at the joint and modal force can form a new vector of the generalized force vector as QsðN þ1Þ1 ¼ fm vm ðtÞ
fmðN þ1Þ1 ¼ ½k u 0 0 0
T
ð5Þ
where ku is the motor torque constant and vm(t) s the voltage applied to the motor. Therefore, a matrix equation for the flexible beam which includes a set of N þ 1 differential equations can therefore be expressed as M s €~q þ H s ~q_ þ K s ~q ¼ Qs
ð6Þ
The mathematical derivation of the mass (Ms), damping (Hs) and stiffness (Ks) matrices for a one-link manipulator can be found in [29]. The mass matrix is
/22 ðrÞ dr .. .
.. .
qAcs
qAcs
RL 0
RL 0
/N ðrÞr dr 0 .. . .. . /2N ðrÞ dr
3 7 7 7 7 7 7 7 7 7 5
ð7Þ
ðN þ1ÞðN þ1Þ
The parameter Jh is the motor-fixture inertia, q is the density of the beam’s material and Acs is the cross-sectional area of the beam. The stiffness matrix is given as
0 0
.. .
0 .. .
Eb I cs
ð4Þ
The parameter fm is defined as
ð3Þ
X
w (r, t)
0
T ~ qðN þ1Þ1 ¼ ½hq1 q2 qN
L
r
0
θ (t)
/2 ðrÞr dr
The joint angle h and modal coordinates qi can form a new vector ~ q of the system’s generalized coordinates as
2
w( r
Z
In this work the flexible beam is modeled using the finite element method, where a continuous structure is discretized into a finite number of elements. The flexible beam has distributed stiffness and mass properties and is regarded as a two-dimensional assemblage modeled as a Euler–Bernoulli beam. For simplicity, the motion of the beam is assumed to take place in the horizontal plane only. Moreover, the motion of a typical point on the structure can be regarded as a superposition of the motion of the joint and the elastic displacement of the point relative to its base. It is assumed that the elastic member does not deform axially. The schematic of the beam system is shown in Fig. 1. The flexible link displacement w at any axial location r can be written in terms of modal coordinates qi and mode shapes /i : N X
313
RL 0
3 7 7 7 7 7 7 7 5
ð8Þ
2
ð/00N ðrÞÞ dr
ðN þ1ÞðN þ1Þ
314
M. Hassan et al. / Mechatronics 17 (2007) 311–323
Ics is the cross-sectional moment of inertia and Eb is the Young’s modulus. In addition, the effect of motor viscous friction and the link structural damping can form a modal damping matrix Hs as 2 6 6 6 6 Hs ¼ 6 6 6 4
0
3
0 0 2f1 m22 x1 0 0 2f2 m33 x2 .. .. .. .. . . . .
0 0 .. .
7 7 7 7 7 7 7 5
0
Bm
0
0
0
2fN mðN þ1ÞðN þ1Þ xN
0
F nPi ¼ fPin vnp ðtÞ
ðN þ1ÞðN þ1Þ
ð9Þ
where Bm is the motor viscous friction coefficient, xi are the system’s natural frequencies, mii are the corresponding elements of the mass matrix in Eq. (7) and fi are the modal damping coefficients. The model in Eq. (6) does not include the effects of the piezoelectric films, which will be derived in the following section. 2.1. Effect of piezoelectric films Piezoelectric film could be used as an actuator or a sensor by applying a voltage or measuring the open circuit voltage, respectively. If the piezoelectric film in Fig. 2 is used as a sensor, the open circuit voltage of the nth PVDF patch V noc ðtÞ is given by Vaz [28] as V noc ðtÞ ¼
N X
WnVi qi ðtÞ
ð10Þ
i¼1
th
where tc is the thickness of the piezoelectric film. Clearly, K npij is zero when i does not equal j due to the orthogonality characteristics of mode shapes. F nPi is the generalized force associated with ith mode. The coefficient of the generalized force is defined as ! Z rne d 31 Ec W c tb d2 /i ðxÞ n n Nc ðxÞ dx ð14Þ fPi ¼ sgnðzc Þ dx2 2 rns where vnp ðtÞ is the voltage applied to the nth PZT actuator. Introducing the effect of piezoelectric films, the new system dynamic model can be described as M ps €~q þ H ps ~q_ þ K ps ~q ¼ fps u
ð15Þ
where the modal mass matrix Mps and damping matrix Hps are as same as Ms and Hs in Eq. (6). The new modal stiffness matrix Kps is a combination of Ks in Eq. (6) and KPij: 2
0 0 0 R L 00 6 2 6 0 Eb I cs 0 ð/ ðrÞÞ dr þ K P 11 0 6 6 .. 6 K ps ¼ 6 0 0 . 6 6. .. .. . . 6 .. . . . 4 0
0
0 Eb I cs
3
0
7 7 7 7 7 7 7 7 7 5
0
RL 0
0 .. . ð/00 ðrÞÞ2 dr þ K PNN
PN PZT
ð11Þ
The variable d31 is the transverse piezoelectric charge to stress ratio, Ec is the Young’s modulus of the film, Cc is the capacitance of the piezoelectric film, Nnc ðxÞ is the shape function of the nth PVDF film, Wc is the maximum width of the piezoelectric film, tb is the thickness of the beam and zc is the coordinate from the neutral axis of the beam to the piezoelectric film. If the piezoelectric film in Fig. 2 is used as an actuator, its effect on the dynamic model is through the passive stiffness and the force produced by the actuator. The contribu-
n
ð13Þ
ð16Þ
where the ith modal coefficient is Z n d 31 Ec W c tb re n d2 /i ðxÞ WnVi ¼ sgnðzc Þ Nc ðxÞ dx dx2 2C c rns
Y
tions of the nth PZT film to the overall stiffness matrix and global force matrix are given by [28]: Z n 2 Ec W c tc t2b re d2 /i ðxÞ d /j ðxÞ dx ð12Þ K nPij ¼ dx2 dx2 4 rns
piezoelectric film
r en r sn
where K Pij ¼ n¼1 K nPij and K nPij is obtained by evaluating Eq. (12) with the parameters of nth PZT film. The parameter NPZT is the number of PZT actuators on the flexible beam. The modal force coefficient matrix fps is 3 2 0 0 ku 0 6 0 f 1 f 2 f N PZT 7 7 6 P1 P1 P1 7 6 6 0 fP12 fP22 fPN2PZT 7 fps ¼ 6 7 ð17Þ 6 . .. 7 .. .. .. 7 6 . . 5 . . . 4 . N PZT 1 2 0 fPN fPN fPN ðN þ1ÞðN PZT þ1Þ where fPin is obtained by evaluating Eq. (14) with the parameters of nth PZT film associated with mode i. The input voltage vector u to the motor that rotates the beam and the voltage to the PZT actuators are: T u ¼ vm ðtÞ v1p ðtÞ v2p ðtÞ vNp PZT ðtÞ ðN þ1Þ1 ð18Þ PZT
tb Wc
wb
r Z tc
L
zc
Fig. 2. Side view of piezoelectric film bonded to a flat beam.
Therefore, the system’s generalized force matrix now includes two components: the force produced by the voltage applied to the motor vm ðtÞ and the force produced by the voltage applied to the piezoelectric actuators vnp ðtÞ (n ¼ 1; 2; 3; ; N PZT ). From the dynamic model in Eq.
M. Hassan et al. / Mechatronics 17 (2007) 311–323
315
(15), the state-space form of the system which is convenient for control simulations can be expressed as
3. Controller formulation
f_xg ¼ ½As fxg þ ½Bs fug;
3.1. Open-loop testing and process prediction
fyg ¼ ½C s fxg
ð19Þ
where the state vector x includes the system’s coordinates ~q and ~ q_ as T x ¼ ½~ q ~ q_ 2ðN þ1Þ1
ð20Þ
The system input vector u includes the voltage applied to motor vm ðtÞ and the voltage applied to the piezoelectric actuators vnp ðtÞ as indicated in Eq. (18). Also, the system output vector includes the potentiometer angular position voltage signal Vh and the piezoelectric sensors’ open circuit voltage V noc : y ¼ ½V h V 1oc V 2oc V nocs Tðns þ1Þ1
ð21Þ
where ns is the total number of PVDF sensors for control simulations. The matrices ½As and ½Bs are defined in terms of the modal stiffness, mass, damping and force coefficient matrices ½K ps , ½M ps , ½H ps and ½fps respectively, and are expressed as " # ½0 ½I As ¼ ð22Þ 1 1 ½½M ps ½K ps ½½M ps ½H ps 2ðN þ1Þ2ðN þ1Þ " # ½0 Bs ¼ ð23Þ ½M ps 1 ½fps 2ðN þ1ÞðN þ1Þ PZT
The matrix ½C s including the potentiometer encoder gain k h and the coefficients of the open circuit voltage for ns PVDF patches is expressed as 3 22 3 0 0 0 kh 7 66 7 7 6 6 0 W1V 1 W1V 2 W1VN 7 7 66 7 7 66 7 7 6 6 0 W2V 1 W2V 2 W2VN 7 Cs ¼ 6 6 7 ½0 7 7 66 7 7 66 .. .. 7 .. 7 66 0 7 . . . 5 44 5 ns ns ns 0 WV 1 WV 2 WVN ðns þ1Þ2ðN þ1Þ ð24Þ Wnvi
where is the nth PVDF patch coefficient of the open circuit voltage for the ith mode and is obtained by evaluating Eq. (11) with the parameters of the PVDF patches.
Model-based predictive control (MPC) is based on the principle of minimizing an objective function J that contains a vector of future errors E over a prediction horizon P, resulting in changes on nu control actions every sampling instant. The value of prediction horizon P is based on the number of discrete sampling intervals required to reach within 95% of the plant output steady-state when excited by a change in the manipulated variable [23]. The changes in the manipulated variable are used to evaluate a prediction of the controlled variable which is the beam vibration at discrete sampling instants. The error vector E is evaluated as the difference between a prediction ^y of the vibration and a setpoint trajectory ysp over P. The parameter ysp is to achieve a zero steady-state. The general MPC architecture can be found in [26] and illustrated in Fig. 3. MPC uses step (or impulse) response values ai obtained by conducting open-loop tests on the flexible beam and the DC motor, formulated into a dynamic matrix Aij. A single dynamic matrix Aij has the structure as 2 3 0 0 0 a1 6a a1 0 0 7 6 2 7 Aij ¼ 6 ð25Þ .. .. 7 .. 6 .. 7 4 . . . 5 . aP
aP 1
aP 2
aP nuþ1
In this study, i is defined as the controlled variable number and j is the manipulated variable number. The manipulated variable and the controlled variable are termed interchangeably as the control move and the plant or process output correspondingly. There are two inputs to the system, one to the DC motor vm and one to the piezoelectric actuators vp. There are two controlled variables for the overall system. One is the motor or joint angular position h which is measured by a high precision potentiometer that provides an analog voltage signal Vh. The second is the beam vibration which is measured by a strain gauge sensor that provides an analog voltage output Voc. The dynamic matrix A of the predictive controller in control simulations and experiments are made up by different
disturbance constraints reference trajectories
future errors
optimizer
P nu
future inputs
cost function model predictor adjustment
Fig. 3. General MPC architecture.
Process
Predicted output
controlled variables
316
M. Hassan et al. / Mechatronics 17 (2007) 311–323
sub-matrices. There are four sub-matrices A11, A12, A21, A22 which contain open-loop response data. A11 relates the motor control voltage vm to joint angle Vh, A21 relates vm to strain gauge voltage Voc, A12 relates the piezoelectric actuator control voltage vp to joint angle Vh, A22 relates the vp to Voc. A12 is almost a zero matrix due to the negligible effect of the piezoelectric actuator control voltage vp on the potentiometer output Vh. The MIMO matrix A is A11 A12 A¼ ð26Þ A21 A22 Let the element of the submatrix Aij in z-transform be expressed as " #T aij1 z1 aij2 z2 aijP zP ð27Þ 0 aij1 z1 aijP 1 zP þ1 Eq. (27) can be defined as Aij ðz1 Þ and hence the discrete form of A can be also expressed as Aðz1 Þ. From these coefficients, a convolution model for the process output or controlled variable that incorporates the effect of the past control move changes is yðzÞ ¼ y 0 þ
P X
i
1
T
Aðz Þ½ð1 z ÞuðzÞ þ DðzÞ ¼ ½ y 1 ðzÞ y 2 ðzÞ
i¼1
ð28Þ where y 1 ðzÞ and y 2 ðzÞ represent the motor output and tip displacement respectively. The term DðzÞ lumps the unmeasurable and measurable disturbances and y0 is the initial condition vector of the measured outputs. Since DðzÞ is time variant, only a prediction of the process can be made. These predictions can be evaluated k time instants ahead of the current instant t using the step response coefficients in A and the control horizon nu. This prediction is the cornerstone of MPC since it provides an insight into the dynamic behavior of the process k steps ahead in the future. Furthermore, the evaluation of future errors over the prediction horizon P can be conducted. A prediction vector ^y k ðzÞ of the process output at the current instant t, k sampling instants ahead based on the past, current, and future control moves, and any nu can be expressed as ^y k ðzÞ ¼ z0 ^y ðzÞ þ
k X
tialized to y0 at the start of control. The first term of Eq. (29) represents the current prediction for the process, the second contains the process prediction due to the present and future control moves, and the third term contains the effect of past control moves. The scalar component dk ðzÞ for each controlled variable is introduced as an adjustment parameter at any instant k that compensates for external disturbances, model mismatch and process nonlinearities. It is important in the determination of ^y k ðzÞ since ^y k ðzÞ is used in the evaluation of the vector of future errors E over P. The evaluation of dk ðzÞ is conducted as time recedes, that is, instant t becomes the new ðt 1Þ and future instant ðt þ 1Þ becomes the current time t. Therefore, dðzÞ over the horizon P at the new current time instant t is determined as the difference between the measured process output at time t and, the process prediction up to time instant t based only on past changes in the manipulated variables. This is expressed by Eq. (30) as dk ðzÞ ¼ y k ðzÞ ^y k ðzÞjz1 with dk ðt þ itÞ ¼ zi dk ðt þ iÞ;
i ¼ 1; 2; . . . ; P
3.2. Projected desired trajectory and predicted errors The MPC strategy focuses on the minimization of future errors contained in E determined from the difference between a setpoint trajectory, ysp, and a predicted process profile, ^y . To avoid an instantaneous step change in the desired setpoint profile from an initial state (which can be zero), a less aggressive setpoint trajectory that would cause the process output to reach in a smooth manner without abrupt control actions can be generated. This profile is called a projected desired trajectory (PDT), s. As a design principle, the discrete setpoint values can belong to a PDT that commences from actual values of the measured adjusted predicted profile y^
i¼knuþ1
þ
kþP 1 X
tðAðzi Þ zk AðzðikÞ ÞÞzk ½ð1 z1 ÞuðzÞ
δ
i¼kþ1 k
þ d ðzÞ 8k; k ¼ 1 . . . P ; i > 0; nu < P
ð29Þ
The parameter yðzÞ contains the control moves to the motor and PZT actuators. Eq. (29) is not commonly found in the literature, and is therefore shown here to generalize the evaluation of the ^y k ðzÞ, k instants ahead for any control horizon nu, for nu < P . The vector term y0 is removed without loss of generality since the initial prediction of the processes are ini-
ð31Þ
The component ^y ðzÞjz1 in Eq. (30) represents the prediction of the process output at instant t made at ðt 1Þ. The value of d as expressed in Eq. (31) is applied to the previous ^y over the receding horizon as illustrated in Fig. 4.
y^
Aðzi Þzk ½ð1 z1 ÞuðzÞ
ð30Þ
^ y(z)| z-1
current predicted profile
t t-1
t+1 t
t+1
current time scale future time scale
Fig. 4. Predicted profile adjustment with time receding.
M. Hassan et al. / Mechatronics 17 (2007) 311–323
process output yðzÞ every sampling instant. In MPC, the formulation of this PDT provides a mechanism to guide the process output trajectory safely, and therefore contributes to reducing the need for computational intensive optimization routines required to suppress the control actions. An efficient way of generating the PDT consists of using the output of a stable model having the setpoint as an input and the process output as initial conditions. This type of response is typical of a second order model with critical damping, which will exhibit the fastest response without overshoot [20]. Another response type for the PDT without overshoot is by means of a trajectory that has first order characteristics, which is generally easier to implement. The PDT, s, for each controlled variable generated over the prediction horizon P from the measured process output yðzÞ at time t for any instant k in the future is given by ski ðzÞ ¼ aki y i ðzÞ þ ð1 ai Þy spi
k X
ah1 ; i
k ¼ 1; 2; . . . ; P
h¼1
ð32Þ where ai is a tunable constant for controlled variable i, 0 6 ai < 1. The superscript on s is the instant k and subsequently onwards for other parameters. The component yspi is fixed and indicates the target value of the PDT as the process reaches its steady-state with minimal error. The PDT is regenerated from the instantaneous measurement of the process y i ðzÞ every sampling instant and is illustrated in Fig. 5. Eq. (32) shows that the PDT is faster in response and approaches a step change in the setpoint or processing condition as ai ) 0.
Unconstrained least square errors (LSE) optimization is the most commonly used error minimization technique in MPC. Hence, the performance index J, to be minimized can be defined as J ¼ ET E
ð33Þ
The determination of the control moves or changes in the manipulated variable, Du0 and Du1 that will be sent to the process or plant so that J in Eq. (33) is minimized by oJ oJ solving oDu ¼ 0 and oDu ¼ 0, yielding: 0 1 1
Du ¼ ðAT AÞ AT ðs ^y Þ T
Du ¼ ½Du0 Du1 nu1 The major drawback of the DMC computation given by Eq. (34) is that initially, large changes in the manipulated variable can occur yielding oscillatory responses. This undesirable phenomenon occurs when the AT A matrix is ill-conditioned or approaches singularity. One simple method of reducing the ill-conditionality of the AT A matrix is performed by multiplying its diagonal elements by a number slightly greater than one [21]. An alternative approach is to modify the standard least squares formulation by adding terms to the performance index in order to introduce weights to the solution vector elements. In other words, the performance index given by Eq. (33) is modified [22], i.e., J ¼ ET QE þ DuT RDu
E ¼ s ð^y þ ADuÞ s New ysp PDT0 PDT1 PDT2 Measured process output y
Control actions t
t+1 t+2
t+3
t+4
Sampling instants
Fig. 5. Projected desired trajectory (PDT) every sampling instant.
ð35Þ
The solution obtained becomes, 1
In order to develop the controller structure, the determination of the vector of future errors as mentioned in Subsection 3.2 is necessary. The future error vector to be minimized is expressed as
ð34Þ
where
Du ¼ ðAT QA þ RÞ AT Qðs ^y Þ
3.3. Error minimization and weighting
317
ð36Þ
where Q and R are positive definite weighting matrices. Note that Eq. (36) includes weighting matrix Q that allows different penalties to be placed on the predicted errors. In this study, Q has unit diagonal terms so that all future errors are equally weighted while the diagonal terms in R are set to the move suppression, R ¼ kI where the scalar quantity k serves as the primary tuning parameter [24,25] known as a move suppression coefficient. Details of the predictive control algorithm can be found in [26]. 4. Simulations Control simulations were conducted on a flexible beam that is solidly coupled to the shaft of a DC motor, as shown in Fig. 1. PVDF and PZT films are used as the sensor and actuators for active damping control. The physical characteristics of the beam and piezoelectric films for simulations are given in Tables 1–3. The motor is rotated and controlled to an angular setpoint using a multivariable predictive controller. During rotation, the beam’s vibration is suppressed using different MPC control schemes until angular rotation of the motor has been completed and continuing until the vibration set point is achieved.
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M. Hassan et al. / Mechatronics 17 (2007) 311–323
Table 1 Beam properties
Table 4 Results of simulations
Material Density q (kg/m3) Young’s modulus Eb (N/m2) Section shape Length L (m) Width Wb (m) Thickness tb (m)
Aluminum 2700 6.9 · 1010 Rectangular 1 0.035 0.0019
Table 2 Motor properties Motor-fixture inertia Jh (kg m2) Friction coeff. Bm (N m/rad) Motor torque constant ku (N m/V) Encoder gain kh (V/rad)
0.14 0.95 0.06 0.3979
Control cases
Max. tip displacement after 15 s (mm)
Damping ratio within 15 s (%)
NC MC PC MP
7.85 2.04 1.71 0.46
1.48 2.59 2.58 3.66
Control performances using different control schemes were studied as shown in Table 4. Figs. 6–13 illustrate simulation results with the prediction horizon P ¼ 10 for Vh, P ¼ 16 for Voc, and the control move horizon nu ¼ 2 for Vh and nu ¼ 3 for Voc. It can be seen from Table 4 and Figs. 6–13 that the vibration is suppressed by motor and/ or piezoelectric actuators effectively while the trajectories
Table 3 Piezoelectric film properties
60
PVDF
PZT
Application Charge constant d31 (C/N) Capacitance Cc (F) Young’s modulus Ec (N/m2) Width Wc (m) Thickness tc (m) Length (m) Shape function Nc
Sensor 23 · 1012 2.7 · 106 2 · 109 0.035 2 · 104 0.5 1
Actuator 175 · 1012 2 · 108 6.5 · 1010 0.025 5 · 104 0.025 1
50
Joint angle (degree)
Material
NC MP
40
30
20
The control schemes discussed here are: 10
0
0
5
10
15
Time (s)
Fig. 6. Joint angle comparison between no vibration control and controlled by motor and actuator together.
60
40
Tip displacement (mm)
(1) NC: only joint angle is controlled. The single-input single-output (SISO) controller uses Vh as the controlled variable while the manipulated variable is vm. The dynamic matrix of the SISO controller is ½A11 . (2) MC: beam vibration is suppressed by motor only. One MIMO controller is used. The dynamic matrix of the MIMO controller with controlled variables Vh and Voc and manipulated variable vm is ½A11 A21 T . (3) PC: beam vibration is suppressed by piezoelectric actuators only and two SISO controllers are used for Vh and Voc. One controller is the same as the NC control scheme and the other controller uses Voc as the controlled variable and vp as the manipulated variable. The dynamic matrices of the two SISO controllers are ½A11 and ½A22 respectively. (4) MP: beam vibration is suppressed by the motor and piezoelectric actuators together. In this case, there are two controllers, one MIMO and one SISO. The MIMO controller is the same as the MC control scheme. The dynamic matrix of the SISO controller with controlled variable Voc and the manipulated variable vp is ½A22 .
20
0
-20
NC MC
-40
-60
0
5
10
15
Time (s)
The sub-matrices A11, A21, A22 are described in Section 3.1.
Fig. 7. Tip displacement comparison between no vibration control and controlled by motor.
M. Hassan et al. / Mechatronics 17 (2007) 311–323
319
14
60
12
Control signal to motor: Vm (V)
Tip displacement (mm)
40
20
0
-20
NC PC
NC MC
10 8 6 4 2
-40
0 -60
0
5
10
-2
15
0
5
Time (s)
10
15
Time (s)
Fig. 8. Tip displacement comparison between no vibration control and controlled by actuator.
Fig. 10. Control signal to motor comparison between no vibration control and controlled by motor.
60
14 12
Control signal to motor: Vm (V)
Tip displacement (mm)
40
20
0
-20 NC MP -40
NC MP
10 8 6 4 2 0
-60
0
5
10
15
Time (s)
Fig. 9. Tip displacement comparison between no vibration control and controlled by motor and actuator together.
of joint angle are similar. The best results are obtained when the motor and piezoelectric actuators are used together as shown in Fig. 9. When the system is controlled by motor and piezoelectric actuators together, the tip displacement is reduced by 94% compared to no vibration control. The damping ratio f is increased by 140% under the same situation. Figs. 10 and 11 illustrate the control signal to motor for control schemes NC, MC and MP. The controllers for MC and MP cases calculate the signals based on vibration and joint angle response together, resulting in an oscillatory signal until it reaches steadystate. Figs. 12 and 13 illustrate the control signal to the actuators for control schemes PC and MP. Both signals are oscillatory, however the MP control signal amplitude decays much faster than the PC case which implies that the MP control scheme reaches its setpoint faster.
-2
0
5
10
15
Time (s)
Fig. 11. Control signal to motor comparison between no vibration control and controlled by motor and actuator together.
5. Real time application 5.1. Experimental setup Experiments were conducted to test the control performance of the MPC controller in suppressing structural vibration. A single-link flexible manipulator was constructed as shown in Fig. 14. The flexible structure comprises of a flexible aluminum beam whose physical properties are summarized in Table 1. A DC motor containing a gear head is coupled to the flexible beam, which provides the required angular position. DC amplifiers are used to amplify the manipulated variables. A potentiometer is installed to measure the angular rotation of the beam with the corresponding analog signal as Vh.
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Four 25 mm · 25 mm · 0.5 mm BM500 PZT films connected in a parallel circuit loop are used as the actuators. The two pairs of films are bonded near the root of the beam, 0.075 m and 0.15 m from the coupled end respectively. Each pair of PZT films is bonded on opposite sides of the beam to produce higher strains. The manipulated signal vp is evaluated by the MPC controller and sent to the DC amplifier. A strain gauge is used as the sensor in all the experiments for measuring beam vibration. A 16bit data acquisition (DAQ) board connected to a shielded connector block was used for acquiring Voc and Vh and for sending the control actions vp and vm.
60
Control signal to actuator: Vp (V)
40
20
0
PC
-20
-40
5.2. Experimental results -60
0
5
10
15
Time (s)
Fig. 12. Control signal to actuator for vibration controlled by actuator.
60
Control signal to actuator: Vp (V)
40
20
0 MP -20
-40
-60
0
5
10
15
Time (s)
Fig. 13. Control signal to actuator for vibration controlled by motor and actuator together.
In this section experiments are conducted by rotating the beam to a prescribed joint angle. Beam vibration is suppressed by using different control schemes used in the control simulations in Section 4. The parameters of the MPC controllers are prediction horizon P ¼ 30 for Vh, P ¼ 15 for Voc, and control move horizon nu ¼ 3 for Vh and nu ¼ 2 for Voc. Control performances are shown in Table 5 and the results are shown in Figs. 15 and 16. The experimental results were found to be similar as those obtained from the simulations. Again, the best results are obtained when the motor and piezoelectric actuators are used together as shown in Fig. 15. The tip displacements using MP control scheme were reduced by 88% when compared to NC control scheme. Under the same situation, the average damping ratio f is increased by 76%. The results in Fig. 15 for the MP case indicate a significant reduction in tip displacement to 9 mm as compared to the NC case of 33 mm in less than 1.5 s. Also, the tip displacement is 3 mm within 3 s as compared to the NC case of 18 mm. These results demonstrate that the MIMO predictive controller performs well in suppressing the beam’s vibration within a short time duration. It should be noted that an exact agreement between experiments and simulation results was not achieved. This is attributed to the existence of gear backlash and the joint compliance which were not included in the theoretical model. However, the control performances in term of the average active damping or maximum tip displacement after 15 s are in good agreement. A comparison of NC case between simulation and experiment is shown in Fig. 17.
Table 5 Results of experiments
Fig. 14. Experimental setup.
Control cases
Max. tip displacement after 15 s (mm)
Average damping ratio within 15 s (%)
NC MC PC MP
5 2 2 0.6
1.94 2.39 2.53 3.42
M. Hassan et al. / Mechatronics 17 (2007) 311–323
321
40 30
NC MP
Tip displacement (mm)
20 10 0 -10 -20 -30 -40 -50 0
2
4
6
8
10
12
14
Time (s)
Fig. 15. Tip displacement comparison between no vibration control and controlled by motor and actuator together.
40 30
NC PC
Tip displacement (mm)
20 10 0 -10 -20 -30 -40 -50 0
2
4
6
8
10
12
14
Time (s)
Fig. 16. Tip displacement comparison between no vibration control and controlled by actuator.
60 Experiment Simulation 40
20
0
-20
-40
-60
0
5
10
15
Fig. 17. Tip displacement comparison of no vibration control (NC) case between experimental and simulation results.
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M. Hassan et al. / Mechatronics 17 (2007) 311–323 NC
MC
PC
MP
damping ratio (%)
3.5 3 2.5 2 1.5 40
50
60
70
angular velocity (deg/s)
Fig. 18. The damping ratios of four different control schemes with different angular velocities.
5.3. Damping ratios with different angular velocities The damping ratios of the four different control schemes with different angular velocities were investigated and the results are shown in Fig. 18. Note the damping ratio is the average value of five tests at each angular velocity. It is observed that the MPC controller works well in actively damping the structural vibration for different angular velocities, with the best results obtained when using the MP control scheme. It is also observed that the damping ratios for NC and PC schemes are more sensitive to varying angular velocities. The smallest damping ratio occurs at the slowest angular velocity and the maximum value occurs at angular velocity of 57/s . However, for the MC and MP schemes, the damping ratio is relatively consistent over the range of angular velocities. 6. Conclusions The development of a multivariable predictive controller is unique in its application to control vibration in a flexible structure. The controller demonstrated good control performance when using the combination of piezoelectric actuators and strain gauge sensing, suppressing tip displacement of the beam to less than 1 mm with 15 s. Although no comparisons to other controllers were conducted, the superiority of this method of control over other controllers is that it is capable of predicting the beam tip displacement in the future due to multiple control actions in the past over a time window. This makes the evaluation of these manipulated variables as optimal as possible in comparison to other control strategies commonly used for vibration suppression control. An additional advantage of using a predictive controller is that adjustments can be made on the prediction of beam vibration that takes into account the effects due to nonlinearities in the system. This study successfully demonstrates that predictive control can be applied to suppress vibration on a flexible beam, making it an excellent candidate for future research on topics such as intelligent control using an array of sensors and actuators, as well as controller tuning specifically for other applications such as a multi-jointed flexible structures.
References [1] Sunar M, Rao SS. Recent advances in sensing and control of flexible structures as a piezoelectric materials. Appl Mech Rev 1999;52(10): 1–16. [2] Plump JM, Hubbard Jr JE. Modeling of an active constrained layer damper proc. In: 12th intl congress on acoustics, Toronto, Canada, 1986; #D4-1. [3] Mead J, Denys J. Passive vibration control. New York: John Wiley and Sons; 1999. [4] Lesieutre G, Lee UA. A finite element for beams having segmented active constrained layers with frequency-dependent viscoelastics. Smart Mater Struct 1996;5(5):615–27. [5] Inman DJ, Lam MJ. Active constrained layer damping treatments. In: 6th International conference on recent advances in structural dynamics, UK, 1997, vol. 1. p. 1–20. [6] Ray M, Baz A. Control of nonlinear vibration of beams using active constrained layer damping. J Vib Control 2001;7(4):539–49. [7] Shen IY. Hybrid damping through intelligent constrained layer treatments. J Vib Acoustics 1994;116(3):341–9. [8] Liao WH, Wang KW. On the active–passive hybrid control actions of active constrained layers. ASME J Vib Acoust 1997;119(4): 563–72. [9] Varadan VV, Lim Y, Varadan VK. Closed loop finite-element modeling of active/passive damping in structural vibration control. Smart Mater Struct 1996;5(5):685–94. [10] Baz A. Optimization of energy dissipation characteristics of active constrained layer damping. Smart Mater Struct 1997;6(3):360–8. [11] Bailey T, Hubbard Jr JE. Distributed piezoelectric-polymer active vibration control of a cantilever beam. J Guidance Control Dynam 1985;8(5):605–11. [12] Choi SB, Lee CH. Force tracking control of a flexible gripper driven by piezoceramic actuators. ASME Trans. J Dynam Syst Measur Control 1997;119:439–46. [13] Baz A, Poh S. Performance of an active control system with piezoelectric actuators. J Vib Control 1988;126(2):327–43. [14] Chonan S, Jiang ZW, Sakuma S. Force control of a miniature gripper driven by piezoceramic bimorph cells. J Adv Automat Technol 1994;6:247–54. [15] Yim W, Singh SN. Variable structure adaptive control of a cantilever beam using piezoelectric actuator. J Vib Control 2000;6: 1029–43. [16] Lin CL, Lee GP, Liu VT. Identification and control of a benchmark flexible structure using piezoelectric actuators and aensors. J Vib Control 2003;9(12):1401–20. [17] Wang S, Yeh H, Roschke PN. Robust control for structural systems with parametric and unstructured uncertainties. J Vib Control 2001;7:753–72. [18] Zheng LA. A robust disturbance rejection method for uncertain flexible mechanical vibrating systems under persistent excitation. J Vib Control 2004;10(3):342–57. [19] Yam LH, Yan YJ. Optimal design of number and locations of actuators in active vibration control of a space truss. Smart Mater Struct 2002;11:496–503. [20] Ogata K. Modern control engineering. 2nd ed. Englewood, NJ: Prentice-Hall; 1990. [21] Cutler CR, Ramaker DL. Dynamic matrix control – a computer control algorithm. In: Proc JACC, San Francisco, CA, 1980. [22] Farrell RJ, Polli A. A comparison of unconstrained dynamic matrix control to conventional feedback control for a first order model. Adv Instrum Control 1990;45(2):1033. [23] Qin JS, Badgwell TA. An overview of industrial model predictive control technology. In: Proceedings of the 5th int conf on chemical process control, AIChE Symposium Series 93, Tahoe City, CA, 1997. 316: 232. [24] Shridhar R, Cooper DJ. A tuning strategy for unconstrained SISO model predictive control. Ind Eng Chem Res 1997;36:729.
M. Hassan et al. / Mechatronics 17 (2007) 311–323 [25] Shridhar R, Cooper DJ. A tuning strategy for multivariable unconstrained model predictive controller. Ind Eng Chem Res 1998;10:4003. [26] Dubay R. Self-optimizing MPC of melt temperature in injection moulding. ISA Trans J Sci Eng Measur Automat 2002;41:81–94. [27] Wang R, Hassan M, Dubay R. Predictive control of flexible structures. In: Flexible automation & intelligent manufacturing conference, Toronto, Ontario, 2004. p. 98–104.
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[28] Vaz A. Composite modeling of flexible structures with bonded piezoelectric film actuators and sensors. IEEE Trans Intrum Measur 1998;24(2):513–20. [29] Bravo R. Vibration control of flexible structure using smart materials. PhD dissertation. McMaster University, Canada, 2000.
Active vibration control of a flexible one-link manipulator using a multivariable predictive controller M. Hassan *, R. Dubay, C. Li, R. Wang Mechanical Engineering Department, University of New Brunswick, Fredericton, Canada Received 13 October 2005; accepted 26 February 2007
Abstract This article presents a new application of model-based predictive controller (MPC) for vibration suppression of a flexible one-link manipulator using piezoceramic actuators. Simulation and experimental studies were conducted to investigate the applicability of the MPC strategy to control vibration of the flexible structure having multiple inputs and multiple outputs (MIMO). The performance of the proposed technique was assessed in terms of level of vibration reduction. The results demonstrated that the proposed predictive control strategy is well suited for multivariable control of vibration suppression on flexible structures. 2007 Elsevier Ltd. All rights reserved. Keywords: Active vibrations control; Mechatronics; Smart structures
1. Introduction The interdisciplinary area of active–passive damping of flexible structures has been covered by a large number of excellent research papers due to its high potential for industrial applications. Over the past two decades, piezoelectric materials for sensing and control of vibration on flexible structures have been well-studied [1], with [2] the first to suggest this idea. For small amplitude vibration on very flexible structures, active approaches lead to lightweight and high performance control systems [1]. Conversely, pure passive techniques using layers of viscoelastic materials are low cost, reliable and can provide good vibration control [3]. This approach is widely used in aerospace, automobile, and aeronautical industries. Several investigations [4–6] have focused on the simultaneous use of piezoelectric (active) and viscoelastic (passive) materials for adaptive and robust vibration control. These methods can employ simultaneous or separate active and
*
Corresponding author. Tel.: +1 506 458 7248; fax: +1 506 453 5025. E-mail address: [email protected] (M. Hassan).
0957-4158/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechatronics.2007.02.004
passive actions. Shen [7,8] used active piezoelectric damping and optical sensors to measure vibration displacements of a beam. Varadan et al. [9] introduced the approach of a piezoelectric sensor bonded next to the actuator. Baz [10] introduced the concept of using two symmetrical bonded sets of segmented piezoelectric sensors and actuators on a flexible beam. Bailey and Hubbard [11] used distributed-parameter control theory and a piezoceramic actuator to actively control vibration on a cantilever beam actively. A sliding mode state feedback controller developed by [12] used a piezoelectric actuator for force tracking of a two-fingered flexible gripper. Other researchers [13,14] studied the effect of the actuators on the host structures for vibration control through modal shape analysis. A variable structure adaptive controller developed by [15] to control contact forces on a cantilever beam used only the output force as feedback, resulting in undesirable chattering. Artificial neural networks (ANN) for identification and state feedback control of flexible structures have been implemented with good preliminary results [16]. Robust control focuses on the ability to have good control performance and stability in the presence of uncertainty in the system model as well as its
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M. Hassan et al. / Mechatronics 17 (2007) 311–323
Nomenclature a d f h k q / x Du Nnc WnVi ai d31 e fm fps fPin kh ku mii ns nu q ~q r s tb tc u v vm vnp w x y y0 ysp b y
tunable constant in PDT adjustment parameter modal damping coefficient joint angle tuning parameter material density mode shapes natural frequency control move vector increment shape function of the nth PVDF film ith modal coefficient of PVDF step (or impulse) response value transverse piezoelectric charge to stress ratio error modal force coefficient for motor modal force coefficient for structure and PZT modal force coefficient of PZT potentiometer encoder gain motor torque constant elements of mass matrix number of PVDF sensors control horizon modal coordinates generalized coordinates axial location projected desired trajectory (PDT) of motor position thickness of the beam thickness of the piezoelectric film control move vector beam velocity control voltage applied to the motor control voltage applied to the nth PZT actuator flexural beam deflection state vector system output vector initial condition of the measured output setpoint trajectory predicted process profile
exogenous inputs, including disturbances and noise. The H1 controller compensates for some of these uncertainties in active vibration control [17]. Recently, [18] developed a robust rejection method using a Kalman filter to estimate the system states under persistent excitation. The investigations in control all lack predictability of the flexible structures’ vibrations and dynamic response to multi-actuators or control actions, disturbances and noise. This prediction can be useful for providing tight and robust control performance, as the control actions can be evaluated using these predictions. A large amount of research has focused on the optimization of sensors/actuators numbers and location, an example being [19]. The use of a pre-
zc zi A Acs As Bm Bs Cc Cs D E Eb Ec F nPi Hs Hps I Ics J Jh Ks Kps K nP L Ms Mps N NPZT P Q Qs R V noc Vh Wb Wc
distance from the neutral axis of the beam to the piezoelectric film backward shift operator in z-transform dynamic matrix cross-sectional area of the beam state matrix motor viscous friction coefficient input matrix capacitance of the piezoelectric film output matrix disturbance vector of future errors Young’s modulus of the beam Young’s modulus of the PZT film generalized nth PZT force for the ith mode modal damping matrix for structure modal damping matrix for structure and PZT unity matrix moment of inertia of the beam controller objective function (or performance index) motor-fixture inertia modal stiffness matrix for structure modal stiffness matrix for structure and PZT stiffness matrix of the nth PZT film length of the beam modal mass matrix for structure modal mass matrix for structure and PZT number of modes number of PZT actuators controller predictive horizon weighting matrix for error vector generalized force vector weighting matrix for control move vector open circuit voltage of nth PVDF patch potentiometer voltage signal width of the beam width of the piezoelectric film
dictive control (PC) strategy can address the optimization of sensors/actuators numbers and location due to its robust and predictive architecture, without the need for extensive computational optimization algorithms. Predictive control has been used on very difficult to control systems that are highly nonlinear having time variant properties [26]. Wang et al. [27] have very recently introduced PC for vibration suppression on a motor driven flexible beam, with very good simulation and practical results. These initial results strongly demonstrate that PC can effectively provide good active suppression vibration on a flexible beam. To the best of the authors knowledge, the use of PC for active vibration suppression is almost non-existent, making it a prime
M. Hassan et al. / Mechatronics 17 (2007) 311–323
area of research to explore, develop and implement on practical systems.
Y
Z θ (t) r, t)
2. Modeling of flexible beam dynamics
wðr; tÞ ¼
/i ðrÞqi ðtÞ
ð1Þ
i¼1
where N is the number of modes. The velocity at any location r on the beam is given by _ _ _ tÞ ¼ hðtÞr vðr; tÞ ¼ hðtÞr þ wðr; þ
N X
/i ðrÞq_ i ðtÞ
ð2Þ
i¼1
where denotes differentiation with respect to t, h is joint angle and rh is the rigid-body displacements of the beam.
2
3
J h þ qAcs3 L 6 R 6 qAcs L / ðrÞr dr 1 6 0 6 RL 6 M s ¼ 6 qAcs /2 ðrÞr dr 0 6 6 .. 6 4 . RL qAcs 0 /N ðrÞr dr
qAcs qAcs
RL 0
RL 0
/1 ðrÞr dr
qAcs
RL 0
/21 ðrÞ dr 0 .. .
qAcs
RL 0
0 60 6 6 6 Ks ¼ 6 0 6. 6. 4. 0
0 RL Eb I cs 0 ð/001 ðrÞÞ2 dr 0 .. . 0
Eb I cs
RL 0
0 0 ð/002 ðrÞÞ2 dr .. . 0
Fig. 1. Schematic of the system.
The torque at the joint and modal force can form a new vector of the generalized force vector as QsðN þ1Þ1 ¼ fm vm ðtÞ
fmðN þ1Þ1 ¼ ½k u 0 0 0
T
ð5Þ
where ku is the motor torque constant and vm(t) s the voltage applied to the motor. Therefore, a matrix equation for the flexible beam which includes a set of N þ 1 differential equations can therefore be expressed as M s €~q þ H s ~q_ þ K s ~q ¼ Qs
ð6Þ
The mathematical derivation of the mass (Ms), damping (Hs) and stiffness (Ks) matrices for a one-link manipulator can be found in [29]. The mass matrix is
/22 ðrÞ dr .. .
.. .
qAcs
qAcs
RL 0
RL 0
/N ðrÞr dr 0 .. . .. . /2N ðrÞ dr
3 7 7 7 7 7 7 7 7 7 5
ð7Þ
ðN þ1ÞðN þ1Þ
The parameter Jh is the motor-fixture inertia, q is the density of the beam’s material and Acs is the cross-sectional area of the beam. The stiffness matrix is given as
0 0
.. .
0 .. .
Eb I cs
ð4Þ
The parameter fm is defined as
ð3Þ
X
w (r, t)
0
T ~ qðN þ1Þ1 ¼ ½hq1 q2 qN
L
r
0
θ (t)
/2 ðrÞr dr
The joint angle h and modal coordinates qi can form a new vector ~ q of the system’s generalized coordinates as
2
w( r
Z
In this work the flexible beam is modeled using the finite element method, where a continuous structure is discretized into a finite number of elements. The flexible beam has distributed stiffness and mass properties and is regarded as a two-dimensional assemblage modeled as a Euler–Bernoulli beam. For simplicity, the motion of the beam is assumed to take place in the horizontal plane only. Moreover, the motion of a typical point on the structure can be regarded as a superposition of the motion of the joint and the elastic displacement of the point relative to its base. It is assumed that the elastic member does not deform axially. The schematic of the beam system is shown in Fig. 1. The flexible link displacement w at any axial location r can be written in terms of modal coordinates qi and mode shapes /i : N X
313
RL 0
3 7 7 7 7 7 7 7 5
ð8Þ
2
ð/00N ðrÞÞ dr
ðN þ1ÞðN þ1Þ
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M. Hassan et al. / Mechatronics 17 (2007) 311–323
Ics is the cross-sectional moment of inertia and Eb is the Young’s modulus. In addition, the effect of motor viscous friction and the link structural damping can form a modal damping matrix Hs as 2 6 6 6 6 Hs ¼ 6 6 6 4
0
3
0 0 2f1 m22 x1 0 0 2f2 m33 x2 .. .. .. .. . . . .
0 0 .. .
7 7 7 7 7 7 7 5
0
Bm
0
0
0
2fN mðN þ1ÞðN þ1Þ xN
0
F nPi ¼ fPin vnp ðtÞ
ðN þ1ÞðN þ1Þ
ð9Þ
where Bm is the motor viscous friction coefficient, xi are the system’s natural frequencies, mii are the corresponding elements of the mass matrix in Eq. (7) and fi are the modal damping coefficients. The model in Eq. (6) does not include the effects of the piezoelectric films, which will be derived in the following section. 2.1. Effect of piezoelectric films Piezoelectric film could be used as an actuator or a sensor by applying a voltage or measuring the open circuit voltage, respectively. If the piezoelectric film in Fig. 2 is used as a sensor, the open circuit voltage of the nth PVDF patch V noc ðtÞ is given by Vaz [28] as V noc ðtÞ ¼
N X
WnVi qi ðtÞ
ð10Þ
i¼1
th
where tc is the thickness of the piezoelectric film. Clearly, K npij is zero when i does not equal j due to the orthogonality characteristics of mode shapes. F nPi is the generalized force associated with ith mode. The coefficient of the generalized force is defined as ! Z rne d 31 Ec W c tb d2 /i ðxÞ n n Nc ðxÞ dx ð14Þ fPi ¼ sgnðzc Þ dx2 2 rns where vnp ðtÞ is the voltage applied to the nth PZT actuator. Introducing the effect of piezoelectric films, the new system dynamic model can be described as M ps €~q þ H ps ~q_ þ K ps ~q ¼ fps u
ð15Þ
where the modal mass matrix Mps and damping matrix Hps are as same as Ms and Hs in Eq. (6). The new modal stiffness matrix Kps is a combination of Ks in Eq. (6) and KPij: 2
0 0 0 R L 00 6 2 6 0 Eb I cs 0 ð/ ðrÞÞ dr þ K P 11 0 6 6 .. 6 K ps ¼ 6 0 0 . 6 6. .. .. . . 6 .. . . . 4 0
0
0 Eb I cs
3
0
7 7 7 7 7 7 7 7 7 5
0
RL 0
0 .. . ð/00 ðrÞÞ2 dr þ K PNN
PN PZT
ð11Þ
The variable d31 is the transverse piezoelectric charge to stress ratio, Ec is the Young’s modulus of the film, Cc is the capacitance of the piezoelectric film, Nnc ðxÞ is the shape function of the nth PVDF film, Wc is the maximum width of the piezoelectric film, tb is the thickness of the beam and zc is the coordinate from the neutral axis of the beam to the piezoelectric film. If the piezoelectric film in Fig. 2 is used as an actuator, its effect on the dynamic model is through the passive stiffness and the force produced by the actuator. The contribu-
n
ð13Þ
ð16Þ
where the ith modal coefficient is Z n d 31 Ec W c tb re n d2 /i ðxÞ WnVi ¼ sgnðzc Þ Nc ðxÞ dx dx2 2C c rns
Y
tions of the nth PZT film to the overall stiffness matrix and global force matrix are given by [28]: Z n 2 Ec W c tc t2b re d2 /i ðxÞ d /j ðxÞ dx ð12Þ K nPij ¼ dx2 dx2 4 rns
piezoelectric film
r en r sn
where K Pij ¼ n¼1 K nPij and K nPij is obtained by evaluating Eq. (12) with the parameters of nth PZT film. The parameter NPZT is the number of PZT actuators on the flexible beam. The modal force coefficient matrix fps is 3 2 0 0 ku 0 6 0 f 1 f 2 f N PZT 7 7 6 P1 P1 P1 7 6 6 0 fP12 fP22 fPN2PZT 7 fps ¼ 6 7 ð17Þ 6 . .. 7 .. .. .. 7 6 . . 5 . . . 4 . N PZT 1 2 0 fPN fPN fPN ðN þ1ÞðN PZT þ1Þ where fPin is obtained by evaluating Eq. (14) with the parameters of nth PZT film associated with mode i. The input voltage vector u to the motor that rotates the beam and the voltage to the PZT actuators are: T u ¼ vm ðtÞ v1p ðtÞ v2p ðtÞ vNp PZT ðtÞ ðN þ1Þ1 ð18Þ PZT
tb Wc
wb
r Z tc
L
zc
Fig. 2. Side view of piezoelectric film bonded to a flat beam.
Therefore, the system’s generalized force matrix now includes two components: the force produced by the voltage applied to the motor vm ðtÞ and the force produced by the voltage applied to the piezoelectric actuators vnp ðtÞ (n ¼ 1; 2; 3; ; N PZT ). From the dynamic model in Eq.
M. Hassan et al. / Mechatronics 17 (2007) 311–323
315
(15), the state-space form of the system which is convenient for control simulations can be expressed as
3. Controller formulation
f_xg ¼ ½As fxg þ ½Bs fug;
3.1. Open-loop testing and process prediction
fyg ¼ ½C s fxg
ð19Þ
where the state vector x includes the system’s coordinates ~q and ~ q_ as T x ¼ ½~ q ~ q_ 2ðN þ1Þ1
ð20Þ
The system input vector u includes the voltage applied to motor vm ðtÞ and the voltage applied to the piezoelectric actuators vnp ðtÞ as indicated in Eq. (18). Also, the system output vector includes the potentiometer angular position voltage signal Vh and the piezoelectric sensors’ open circuit voltage V noc : y ¼ ½V h V 1oc V 2oc V nocs Tðns þ1Þ1
ð21Þ
where ns is the total number of PVDF sensors for control simulations. The matrices ½As and ½Bs are defined in terms of the modal stiffness, mass, damping and force coefficient matrices ½K ps , ½M ps , ½H ps and ½fps respectively, and are expressed as " # ½0 ½I As ¼ ð22Þ 1 1 ½½M ps ½K ps ½½M ps ½H ps 2ðN þ1Þ2ðN þ1Þ " # ½0 Bs ¼ ð23Þ ½M ps 1 ½fps 2ðN þ1ÞðN þ1Þ PZT
The matrix ½C s including the potentiometer encoder gain k h and the coefficients of the open circuit voltage for ns PVDF patches is expressed as 3 22 3 0 0 0 kh 7 66 7 7 6 6 0 W1V 1 W1V 2 W1VN 7 7 66 7 7 66 7 7 6 6 0 W2V 1 W2V 2 W2VN 7 Cs ¼ 6 6 7 ½0 7 7 66 7 7 66 .. .. 7 .. 7 66 0 7 . . . 5 44 5 ns ns ns 0 WV 1 WV 2 WVN ðns þ1Þ2ðN þ1Þ ð24Þ Wnvi
where is the nth PVDF patch coefficient of the open circuit voltage for the ith mode and is obtained by evaluating Eq. (11) with the parameters of the PVDF patches.
Model-based predictive control (MPC) is based on the principle of minimizing an objective function J that contains a vector of future errors E over a prediction horizon P, resulting in changes on nu control actions every sampling instant. The value of prediction horizon P is based on the number of discrete sampling intervals required to reach within 95% of the plant output steady-state when excited by a change in the manipulated variable [23]. The changes in the manipulated variable are used to evaluate a prediction of the controlled variable which is the beam vibration at discrete sampling instants. The error vector E is evaluated as the difference between a prediction ^y of the vibration and a setpoint trajectory ysp over P. The parameter ysp is to achieve a zero steady-state. The general MPC architecture can be found in [26] and illustrated in Fig. 3. MPC uses step (or impulse) response values ai obtained by conducting open-loop tests on the flexible beam and the DC motor, formulated into a dynamic matrix Aij. A single dynamic matrix Aij has the structure as 2 3 0 0 0 a1 6a a1 0 0 7 6 2 7 Aij ¼ 6 ð25Þ .. .. 7 .. 6 .. 7 4 . . . 5 . aP
aP 1
aP 2
aP nuþ1
In this study, i is defined as the controlled variable number and j is the manipulated variable number. The manipulated variable and the controlled variable are termed interchangeably as the control move and the plant or process output correspondingly. There are two inputs to the system, one to the DC motor vm and one to the piezoelectric actuators vp. There are two controlled variables for the overall system. One is the motor or joint angular position h which is measured by a high precision potentiometer that provides an analog voltage signal Vh. The second is the beam vibration which is measured by a strain gauge sensor that provides an analog voltage output Voc. The dynamic matrix A of the predictive controller in control simulations and experiments are made up by different
disturbance constraints reference trajectories
future errors
optimizer
P nu
future inputs
cost function model predictor adjustment
Fig. 3. General MPC architecture.
Process
Predicted output
controlled variables
316
M. Hassan et al. / Mechatronics 17 (2007) 311–323
sub-matrices. There are four sub-matrices A11, A12, A21, A22 which contain open-loop response data. A11 relates the motor control voltage vm to joint angle Vh, A21 relates vm to strain gauge voltage Voc, A12 relates the piezoelectric actuator control voltage vp to joint angle Vh, A22 relates the vp to Voc. A12 is almost a zero matrix due to the negligible effect of the piezoelectric actuator control voltage vp on the potentiometer output Vh. The MIMO matrix A is A11 A12 A¼ ð26Þ A21 A22 Let the element of the submatrix Aij in z-transform be expressed as " #T aij1 z1 aij2 z2 aijP zP ð27Þ 0 aij1 z1 aijP 1 zP þ1 Eq. (27) can be defined as Aij ðz1 Þ and hence the discrete form of A can be also expressed as Aðz1 Þ. From these coefficients, a convolution model for the process output or controlled variable that incorporates the effect of the past control move changes is yðzÞ ¼ y 0 þ
P X
i
1
T
Aðz Þ½ð1 z ÞuðzÞ þ DðzÞ ¼ ½ y 1 ðzÞ y 2 ðzÞ
i¼1
ð28Þ where y 1 ðzÞ and y 2 ðzÞ represent the motor output and tip displacement respectively. The term DðzÞ lumps the unmeasurable and measurable disturbances and y0 is the initial condition vector of the measured outputs. Since DðzÞ is time variant, only a prediction of the process can be made. These predictions can be evaluated k time instants ahead of the current instant t using the step response coefficients in A and the control horizon nu. This prediction is the cornerstone of MPC since it provides an insight into the dynamic behavior of the process k steps ahead in the future. Furthermore, the evaluation of future errors over the prediction horizon P can be conducted. A prediction vector ^y k ðzÞ of the process output at the current instant t, k sampling instants ahead based on the past, current, and future control moves, and any nu can be expressed as ^y k ðzÞ ¼ z0 ^y ðzÞ þ
k X
tialized to y0 at the start of control. The first term of Eq. (29) represents the current prediction for the process, the second contains the process prediction due to the present and future control moves, and the third term contains the effect of past control moves. The scalar component dk ðzÞ for each controlled variable is introduced as an adjustment parameter at any instant k that compensates for external disturbances, model mismatch and process nonlinearities. It is important in the determination of ^y k ðzÞ since ^y k ðzÞ is used in the evaluation of the vector of future errors E over P. The evaluation of dk ðzÞ is conducted as time recedes, that is, instant t becomes the new ðt 1Þ and future instant ðt þ 1Þ becomes the current time t. Therefore, dðzÞ over the horizon P at the new current time instant t is determined as the difference between the measured process output at time t and, the process prediction up to time instant t based only on past changes in the manipulated variables. This is expressed by Eq. (30) as dk ðzÞ ¼ y k ðzÞ ^y k ðzÞjz1 with dk ðt þ itÞ ¼ zi dk ðt þ iÞ;
i ¼ 1; 2; . . . ; P
3.2. Projected desired trajectory and predicted errors The MPC strategy focuses on the minimization of future errors contained in E determined from the difference between a setpoint trajectory, ysp, and a predicted process profile, ^y . To avoid an instantaneous step change in the desired setpoint profile from an initial state (which can be zero), a less aggressive setpoint trajectory that would cause the process output to reach in a smooth manner without abrupt control actions can be generated. This profile is called a projected desired trajectory (PDT), s. As a design principle, the discrete setpoint values can belong to a PDT that commences from actual values of the measured adjusted predicted profile y^
i¼knuþ1
þ
kþP 1 X
tðAðzi Þ zk AðzðikÞ ÞÞzk ½ð1 z1 ÞuðzÞ
δ
i¼kþ1 k
þ d ðzÞ 8k; k ¼ 1 . . . P ; i > 0; nu < P
ð29Þ
The parameter yðzÞ contains the control moves to the motor and PZT actuators. Eq. (29) is not commonly found in the literature, and is therefore shown here to generalize the evaluation of the ^y k ðzÞ, k instants ahead for any control horizon nu, for nu < P . The vector term y0 is removed without loss of generality since the initial prediction of the processes are ini-
ð31Þ
The component ^y ðzÞjz1 in Eq. (30) represents the prediction of the process output at instant t made at ðt 1Þ. The value of d as expressed in Eq. (31) is applied to the previous ^y over the receding horizon as illustrated in Fig. 4.
y^
Aðzi Þzk ½ð1 z1 ÞuðzÞ
ð30Þ
^ y(z)| z-1
current predicted profile
t t-1
t+1 t
t+1
current time scale future time scale
Fig. 4. Predicted profile adjustment with time receding.
M. Hassan et al. / Mechatronics 17 (2007) 311–323
process output yðzÞ every sampling instant. In MPC, the formulation of this PDT provides a mechanism to guide the process output trajectory safely, and therefore contributes to reducing the need for computational intensive optimization routines required to suppress the control actions. An efficient way of generating the PDT consists of using the output of a stable model having the setpoint as an input and the process output as initial conditions. This type of response is typical of a second order model with critical damping, which will exhibit the fastest response without overshoot [20]. Another response type for the PDT without overshoot is by means of a trajectory that has first order characteristics, which is generally easier to implement. The PDT, s, for each controlled variable generated over the prediction horizon P from the measured process output yðzÞ at time t for any instant k in the future is given by ski ðzÞ ¼ aki y i ðzÞ þ ð1 ai Þy spi
k X
ah1 ; i
k ¼ 1; 2; . . . ; P
h¼1
ð32Þ where ai is a tunable constant for controlled variable i, 0 6 ai < 1. The superscript on s is the instant k and subsequently onwards for other parameters. The component yspi is fixed and indicates the target value of the PDT as the process reaches its steady-state with minimal error. The PDT is regenerated from the instantaneous measurement of the process y i ðzÞ every sampling instant and is illustrated in Fig. 5. Eq. (32) shows that the PDT is faster in response and approaches a step change in the setpoint or processing condition as ai ) 0.
Unconstrained least square errors (LSE) optimization is the most commonly used error minimization technique in MPC. Hence, the performance index J, to be minimized can be defined as J ¼ ET E
ð33Þ
The determination of the control moves or changes in the manipulated variable, Du0 and Du1 that will be sent to the process or plant so that J in Eq. (33) is minimized by oJ oJ solving oDu ¼ 0 and oDu ¼ 0, yielding: 0 1 1
Du ¼ ðAT AÞ AT ðs ^y Þ T
Du ¼ ½Du0 Du1 nu1 The major drawback of the DMC computation given by Eq. (34) is that initially, large changes in the manipulated variable can occur yielding oscillatory responses. This undesirable phenomenon occurs when the AT A matrix is ill-conditioned or approaches singularity. One simple method of reducing the ill-conditionality of the AT A matrix is performed by multiplying its diagonal elements by a number slightly greater than one [21]. An alternative approach is to modify the standard least squares formulation by adding terms to the performance index in order to introduce weights to the solution vector elements. In other words, the performance index given by Eq. (33) is modified [22], i.e., J ¼ ET QE þ DuT RDu
E ¼ s ð^y þ ADuÞ s New ysp PDT0 PDT1 PDT2 Measured process output y
Control actions t
t+1 t+2
t+3
t+4
Sampling instants
Fig. 5. Projected desired trajectory (PDT) every sampling instant.
ð35Þ
The solution obtained becomes, 1
In order to develop the controller structure, the determination of the vector of future errors as mentioned in Subsection 3.2 is necessary. The future error vector to be minimized is expressed as
ð34Þ
where
Du ¼ ðAT QA þ RÞ AT Qðs ^y Þ
3.3. Error minimization and weighting
317
ð36Þ
where Q and R are positive definite weighting matrices. Note that Eq. (36) includes weighting matrix Q that allows different penalties to be placed on the predicted errors. In this study, Q has unit diagonal terms so that all future errors are equally weighted while the diagonal terms in R are set to the move suppression, R ¼ kI where the scalar quantity k serves as the primary tuning parameter [24,25] known as a move suppression coefficient. Details of the predictive control algorithm can be found in [26]. 4. Simulations Control simulations were conducted on a flexible beam that is solidly coupled to the shaft of a DC motor, as shown in Fig. 1. PVDF and PZT films are used as the sensor and actuators for active damping control. The physical characteristics of the beam and piezoelectric films for simulations are given in Tables 1–3. The motor is rotated and controlled to an angular setpoint using a multivariable predictive controller. During rotation, the beam’s vibration is suppressed using different MPC control schemes until angular rotation of the motor has been completed and continuing until the vibration set point is achieved.
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M. Hassan et al. / Mechatronics 17 (2007) 311–323
Table 1 Beam properties
Table 4 Results of simulations
Material Density q (kg/m3) Young’s modulus Eb (N/m2) Section shape Length L (m) Width Wb (m) Thickness tb (m)
Aluminum 2700 6.9 · 1010 Rectangular 1 0.035 0.0019
Table 2 Motor properties Motor-fixture inertia Jh (kg m2) Friction coeff. Bm (N m/rad) Motor torque constant ku (N m/V) Encoder gain kh (V/rad)
0.14 0.95 0.06 0.3979
Control cases
Max. tip displacement after 15 s (mm)
Damping ratio within 15 s (%)
NC MC PC MP
7.85 2.04 1.71 0.46
1.48 2.59 2.58 3.66
Control performances using different control schemes were studied as shown in Table 4. Figs. 6–13 illustrate simulation results with the prediction horizon P ¼ 10 for Vh, P ¼ 16 for Voc, and the control move horizon nu ¼ 2 for Vh and nu ¼ 3 for Voc. It can be seen from Table 4 and Figs. 6–13 that the vibration is suppressed by motor and/ or piezoelectric actuators effectively while the trajectories
Table 3 Piezoelectric film properties
60
PVDF
PZT
Application Charge constant d31 (C/N) Capacitance Cc (F) Young’s modulus Ec (N/m2) Width Wc (m) Thickness tc (m) Length (m) Shape function Nc
Sensor 23 · 1012 2.7 · 106 2 · 109 0.035 2 · 104 0.5 1
Actuator 175 · 1012 2 · 108 6.5 · 1010 0.025 5 · 104 0.025 1
50
Joint angle (degree)
Material
NC MP
40
30
20
The control schemes discussed here are: 10
0
0
5
10
15
Time (s)
Fig. 6. Joint angle comparison between no vibration control and controlled by motor and actuator together.
60
40
Tip displacement (mm)
(1) NC: only joint angle is controlled. The single-input single-output (SISO) controller uses Vh as the controlled variable while the manipulated variable is vm. The dynamic matrix of the SISO controller is ½A11 . (2) MC: beam vibration is suppressed by motor only. One MIMO controller is used. The dynamic matrix of the MIMO controller with controlled variables Vh and Voc and manipulated variable vm is ½A11 A21 T . (3) PC: beam vibration is suppressed by piezoelectric actuators only and two SISO controllers are used for Vh and Voc. One controller is the same as the NC control scheme and the other controller uses Voc as the controlled variable and vp as the manipulated variable. The dynamic matrices of the two SISO controllers are ½A11 and ½A22 respectively. (4) MP: beam vibration is suppressed by the motor and piezoelectric actuators together. In this case, there are two controllers, one MIMO and one SISO. The MIMO controller is the same as the MC control scheme. The dynamic matrix of the SISO controller with controlled variable Voc and the manipulated variable vp is ½A22 .
20
0
-20
NC MC
-40
-60
0
5
10
15
Time (s)
The sub-matrices A11, A21, A22 are described in Section 3.1.
Fig. 7. Tip displacement comparison between no vibration control and controlled by motor.
M. Hassan et al. / Mechatronics 17 (2007) 311–323
319
14
60
12
Control signal to motor: Vm (V)
Tip displacement (mm)
40
20
0
-20
NC PC
NC MC
10 8 6 4 2
-40
0 -60
0
5
10
-2
15
0
5
Time (s)
10
15
Time (s)
Fig. 8. Tip displacement comparison between no vibration control and controlled by actuator.
Fig. 10. Control signal to motor comparison between no vibration control and controlled by motor.
60
14 12
Control signal to motor: Vm (V)
Tip displacement (mm)
40
20
0
-20 NC MP -40
NC MP
10 8 6 4 2 0
-60
0
5
10
15
Time (s)
Fig. 9. Tip displacement comparison between no vibration control and controlled by motor and actuator together.
of joint angle are similar. The best results are obtained when the motor and piezoelectric actuators are used together as shown in Fig. 9. When the system is controlled by motor and piezoelectric actuators together, the tip displacement is reduced by 94% compared to no vibration control. The damping ratio f is increased by 140% under the same situation. Figs. 10 and 11 illustrate the control signal to motor for control schemes NC, MC and MP. The controllers for MC and MP cases calculate the signals based on vibration and joint angle response together, resulting in an oscillatory signal until it reaches steadystate. Figs. 12 and 13 illustrate the control signal to the actuators for control schemes PC and MP. Both signals are oscillatory, however the MP control signal amplitude decays much faster than the PC case which implies that the MP control scheme reaches its setpoint faster.
-2
0
5
10
15
Time (s)
Fig. 11. Control signal to motor comparison between no vibration control and controlled by motor and actuator together.
5. Real time application 5.1. Experimental setup Experiments were conducted to test the control performance of the MPC controller in suppressing structural vibration. A single-link flexible manipulator was constructed as shown in Fig. 14. The flexible structure comprises of a flexible aluminum beam whose physical properties are summarized in Table 1. A DC motor containing a gear head is coupled to the flexible beam, which provides the required angular position. DC amplifiers are used to amplify the manipulated variables. A potentiometer is installed to measure the angular rotation of the beam with the corresponding analog signal as Vh.
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M. Hassan et al. / Mechatronics 17 (2007) 311–323
Four 25 mm · 25 mm · 0.5 mm BM500 PZT films connected in a parallel circuit loop are used as the actuators. The two pairs of films are bonded near the root of the beam, 0.075 m and 0.15 m from the coupled end respectively. Each pair of PZT films is bonded on opposite sides of the beam to produce higher strains. The manipulated signal vp is evaluated by the MPC controller and sent to the DC amplifier. A strain gauge is used as the sensor in all the experiments for measuring beam vibration. A 16bit data acquisition (DAQ) board connected to a shielded connector block was used for acquiring Voc and Vh and for sending the control actions vp and vm.
60
Control signal to actuator: Vp (V)
40
20
0
PC
-20
-40
5.2. Experimental results -60
0
5
10
15
Time (s)
Fig. 12. Control signal to actuator for vibration controlled by actuator.
60
Control signal to actuator: Vp (V)
40
20
0 MP -20
-40
-60
0
5
10
15
Time (s)
Fig. 13. Control signal to actuator for vibration controlled by motor and actuator together.
In this section experiments are conducted by rotating the beam to a prescribed joint angle. Beam vibration is suppressed by using different control schemes used in the control simulations in Section 4. The parameters of the MPC controllers are prediction horizon P ¼ 30 for Vh, P ¼ 15 for Voc, and control move horizon nu ¼ 3 for Vh and nu ¼ 2 for Voc. Control performances are shown in Table 5 and the results are shown in Figs. 15 and 16. The experimental results were found to be similar as those obtained from the simulations. Again, the best results are obtained when the motor and piezoelectric actuators are used together as shown in Fig. 15. The tip displacements using MP control scheme were reduced by 88% when compared to NC control scheme. Under the same situation, the average damping ratio f is increased by 76%. The results in Fig. 15 for the MP case indicate a significant reduction in tip displacement to 9 mm as compared to the NC case of 33 mm in less than 1.5 s. Also, the tip displacement is 3 mm within 3 s as compared to the NC case of 18 mm. These results demonstrate that the MIMO predictive controller performs well in suppressing the beam’s vibration within a short time duration. It should be noted that an exact agreement between experiments and simulation results was not achieved. This is attributed to the existence of gear backlash and the joint compliance which were not included in the theoretical model. However, the control performances in term of the average active damping or maximum tip displacement after 15 s are in good agreement. A comparison of NC case between simulation and experiment is shown in Fig. 17.
Table 5 Results of experiments
Fig. 14. Experimental setup.
Control cases
Max. tip displacement after 15 s (mm)
Average damping ratio within 15 s (%)
NC MC PC MP
5 2 2 0.6
1.94 2.39 2.53 3.42
M. Hassan et al. / Mechatronics 17 (2007) 311–323
321
40 30
NC MP
Tip displacement (mm)
20 10 0 -10 -20 -30 -40 -50 0
2
4
6
8
10
12
14
Time (s)
Fig. 15. Tip displacement comparison between no vibration control and controlled by motor and actuator together.
40 30
NC PC
Tip displacement (mm)
20 10 0 -10 -20 -30 -40 -50 0
2
4
6
8
10
12
14
Time (s)
Fig. 16. Tip displacement comparison between no vibration control and controlled by actuator.
60 Experiment Simulation 40
20
0
-20
-40
-60
0
5
10
15
Fig. 17. Tip displacement comparison of no vibration control (NC) case between experimental and simulation results.
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M. Hassan et al. / Mechatronics 17 (2007) 311–323 NC
MC
PC
MP
damping ratio (%)
3.5 3 2.5 2 1.5 40
50
60
70
angular velocity (deg/s)
Fig. 18. The damping ratios of four different control schemes with different angular velocities.
5.3. Damping ratios with different angular velocities The damping ratios of the four different control schemes with different angular velocities were investigated and the results are shown in Fig. 18. Note the damping ratio is the average value of five tests at each angular velocity. It is observed that the MPC controller works well in actively damping the structural vibration for different angular velocities, with the best results obtained when using the MP control scheme. It is also observed that the damping ratios for NC and PC schemes are more sensitive to varying angular velocities. The smallest damping ratio occurs at the slowest angular velocity and the maximum value occurs at angular velocity of 57/s . However, for the MC and MP schemes, the damping ratio is relatively consistent over the range of angular velocities. 6. Conclusions The development of a multivariable predictive controller is unique in its application to control vibration in a flexible structure. The controller demonstrated good control performance when using the combination of piezoelectric actuators and strain gauge sensing, suppressing tip displacement of the beam to less than 1 mm with 15 s. Although no comparisons to other controllers were conducted, the superiority of this method of control over other controllers is that it is capable of predicting the beam tip displacement in the future due to multiple control actions in the past over a time window. This makes the evaluation of these manipulated variables as optimal as possible in comparison to other control strategies commonly used for vibration suppression control. An additional advantage of using a predictive controller is that adjustments can be made on the prediction of beam vibration that takes into account the effects due to nonlinearities in the system. This study successfully demonstrates that predictive control can be applied to suppress vibration on a flexible beam, making it an excellent candidate for future research on topics such as intelligent control using an array of sensors and actuators, as well as controller tuning specifically for other applications such as a multi-jointed flexible structures.
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