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ISME Journal of Mechanics and Design

Vol. 2, No.1, 2018 pp. 1-8

Active Vibration Control of a Plate Using Linear Quadratic Tracking

Control

B. Kheiri Sarabi1* and M. Sharma2

1Maad Energy Spico Co., Khorramabad, Iran 2UIET, Panjab University, Chandigarh, 160014, India.

*Corresponding author: behrooz.kheiri@gmail.com

Abstract In this work, a novel and simple technique is proposed to suppress vibrations in a structure with desired transient decay curves of individual modes of vibration. To illustrate this technique a cantilevered plate instrumented with piezoelectric patches is chosen to be studied. The finite element technique is used to derive a mathematical model of the cantilevered plate and then the finite element model is converted to a state-space model. A Linear Quadratic Tracking (LQT) control technique is employed to suppress vibrations in a plate structure by simultaneously tracking the desired transient decay curves for the first three modes of vibration. Simulation results show that the presented technique for active vibration control allows to precisely dictate transient decay vibration response of an active structure. Keywords: Active vibration control, Linear quadratic tracking, Finite element model, cantilevered plate. 1. Introduction

In the last few decades, technique of vibration control in which some external source of energy is used, has attracted a lot of interest. This technique is called Active Vibration Control (AVC) and it essentially requires a sensor, actuator, processor & external source of energy for its operation [1, 2, 3]. AVC techniques can control structural vibrations without significantly increasing the overall weight of the structure. Feedback and feedforward control laws are generally used as basic control techniques in AVC. General feed forward control technique can be used to control noise as well as structural vibrations [4]. Feedforward technique is in form of adaptive filtering of disturbances for calculation of control signal [5]. In feedback control, signal of the variable to be controlled is used in the control law. Constant gain and constant amplitude control algorithms can be used to control vibrations of a structure [6, 7]. Negative velocity feedback algorithm has been widely used for active control of vibrations [8]. In a direct velocity feedback technique, feedback signal from velocity sensor multiplied with some gain is used for computing force to be applied by an actuator [9]. Positive position feedback (PPF) control technique can also be used as a control technique to control dynamic behavior of structures [10]. Acceleration feedback control technique with finite actuator dynamics can be used in flexible structures to overcome instability [11]. Stability based on the second direct method of Lyapunov feedback can be used to control total mechanical energy of the system [12]. Well known techniques like pole-placement [13], H∞ control [14] etc have also been exploited for usage in AVC. Modal control techniques have obtained wide acceptability in AVC [15, 16]. In these techniques a mathematical model of the structure is reduced to first few frequencies only using modal truncation. In Independent Modal Space Control (IMSC) the entire control effort is spent to control a particular mode of vibration only [17]. On the other hand, in Modified Independent Modal Space Control (MIMSC) at a particular instant of time the entire control effort is spent to control a mode that has the highest energy at that particular instant of time. Therefore MIMSC unlike IMSC is able to control multiple modes [18]. Optimal controls like LQR, LQG etc are based on minimization/maximization of a performance index for calculation

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of control gains. In AVC applications, the performance index is usually a quadratic function of structural vibrations and control voltages. Numerous works have been done to control structural vibrations using optimal control. In all these works, optimal control has been used as a regulator to suppress structural vibrations to the mean position i.e. the zero position [19, 20]. If desired transient decay curves are not given as reference then performance of an active structure may appreciably change if due to some reason there is gain/loss of mass or/and stiffness or/and damping of an active structure. This is a clear gap in the field of AVC which the authors feel should be plugged to make performance of smart structures robust. There is a dire need to have an AVC strategy that enables the designer of active structures to design tailor-made transient response of a smart structure. This work will be of interest when it is desired to maintain a particular level of vibrations during journey of vibration decay so as to meet some functional requirement. There is no work in which optimal control has been used to simultaneously achieve desired transient decay curves of individual modes of vibration. In the present work, optimal tracking control has been used to suppress structural vibrations of a cantilevered plate with desired transient decay curves of individual modes. Cantilevered plate is instrumented with one piezelectric sensor and two piezoelectric actuators. AVC technique presented in this work allows to exercise more control on response of an active structure and allows to precisely dictate the vibratory response of a structure. In section 2, Finite Element Model of plate is derived using Hamilton’s principle. In section 3, Linear Quadratic Tracking (LQT) technique is discussed and developed to track transient references. In section 4, results are presented and in section 5 conclusions are drawn.

2. Finite element model of smart plate Finite element technique has been used extensively to derive mathematical model of active structures [21, 22]. In this paper finite element technique based on Hamilton’s principle has been used to develop mathematical model of a cantilevered plate. Consider a thin cantilevered mild-steel plate, as shown in figure (1). This plate is divided into 64 quadrilateral elements of equal size. Each element has four nodes and each node has three degrees of freedom (one flexural displacement, one rotation about x-axis and one rotation about y-axis). One piezoelectric sensor patch is pasted at 10th element and two piezoelectric patches are pasted at 12th & 14th elements as actuators. Piezoelectric sensor and actuators have been placed in these locations as at these locations modal strains of first three modes of vibration are high. Therefore these locations of piezoelectric patches are suitable to sense as well as control first three modes of vibrations.

Hamilton’s principle for an individual finite element in arbitrarily selected time intervals 𝑡0 to 𝑡1 is expressed as:

∫ δ(Te − Ue + We)

t1

t0

dt = 0 (1)

where Te is kinetic energy of one finite element, Ue is potential energy of one finite element, We is sum of energy stored by surface forces in one finite element. Expressions of kinetic energy, potential energy and work done are evaluated for one finite element and substituted in equation (1). Equation of motion of a finite element is derived by taking variation as suggested in equation (1). Equations of motion of individual finite elements are systematically assembled to derive global matrix equation of motion of the entire structure. Matrix equation of motion of the entire structure is expressed as:

[𝑀]{�̈�} + [𝐷]{�̇�} + [𝐾]{𝑑} = {𝐹} (2)

where [M] is mass matrix of structure, [𝐷] is damping matrix of structure, [𝐾] is stiffness matrix of structure, {𝐹} is external force vector applied by piezoelectric actuator and {𝑑} is displacement vector. The

Fig. 1 Cantilevered plate instrumented with piezoelectric patches

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damping matrix has been constructed by assuming it to be viscous damping that is proportional to a linear combination of mass and stiffness matrices of the smart plate. Equation (2) can be written in modal domain by performing orthonormal modal transformation given by {d(t)} = [U]{η(t)}:

[M][U]{η̈(t)} + [K][U]{η(t)} = [D]{fa(t)} + {fe(t)} (3)

Where [U] is orthonormal modal matrix. Pre-multiplying equation (2) by [U]T, equation (3) becomes:

[I]{η̈(t)} + [λr

2]{η(t)} = [U]T[D]{fa(t)} + [U]T{fe(t)} (4)

The equation of motion of rth mode is:

η̈r + λr

2ηr = qr(t) + yr(t) r = 1…n (5)

ηr, qr, yr represent modal displacement, modal control force and modal excitation force respectively of rth mode and λr is rthmode natural frequency. For first three modes equations are derived as:

η̈1 + λ12η1 = q1

η̈2 + λ22η2 = q2

η̈3 + λ32η3 = q3

(6)

The finite element model of structure is reduced to first three modes using modal truncation. Reduced finite element model using modal truncation is converted into state-space model and state-space model is further converted into a discrete state-space model. State-space model of the intelligent plate is expressed as:

�̇�(𝑡) = 𝐴𝑥(𝑡) + 𝐵𝑢(𝑡) (7)

where 𝐴 is system matrix, 𝐵 is control matrix, 𝑥(𝑡) is state vector of system and 𝑢(𝑡) is control input.

3. Discrete-time linear quadratic tracking control Finite element model of the smart plate is discretized using a finite sampling time interval into a discrete state-space model of form:

𝑥(𝑘 + 1) = 𝑆𝑥(𝑘) + 𝐺𝑢(𝑘) (8)

with output relation as:

𝑦(𝑘) = C𝑥(𝑘) (9)

where 𝑦(𝑘) is piezoelectric sensor output vector, 𝑘 is a sampling instant, 𝑆, 𝐺 & 𝐶 are discrete system matrix (𝑛 × 𝑛), control matrix (𝑛 × 1) & sensor vector (1 × 𝑛) respectively. Voltage developed across a piezoelectric sensor is derived as:

vsen = [kvu]Zsen−1 x(k) (10)

where [kvu] is electromechanical interaction matrix of piezoelectric & structure and Zsen is capacitance of piezoelectric sensor. Similarly, a piezoelectric patch is used as an actuator to apply the control voltages. The force applied by a piezoelectric actuator is:

fact = [kvu]Zsen−1 ZactV(k)

(11)

Where Zact is capacitance of piezoelectric actuator. The values of 𝑆 & 𝐺 matrices are depended on system and for present system are computed as:

S =

[

0.99−7.62𝑒−18

−7.72𝑒−18

−14.41−2.27𝑒−14

−2.27𝑒−14

−3.67𝑒−17

0.961.25𝑒−17

−1.09𝑒−13

−82.443.68𝑒−14

−1.55𝑒−16

5.24𝑒−17

0.79−4.60𝑒−13

1.54𝑒−13

−406.36

9.95𝑒−4

1.57𝑒−18

1.57𝑒−18

0.993.11𝑒−15

3.02𝑒−15

1.31𝑒−18

9.83𝑒−4

−4.39𝑒−19

2.59𝑒−15

0.95−8.39𝑒−16

1.05𝑒−18

−3.52𝑒−19

9.27𝑒−4

2.01𝑒−15

−6.72𝑒−16

0.78 ]

(12)

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G =

[ −3.84𝑒−10

1.13𝑒−10

−4.00𝑒−10

−7.66𝑒−7

2.25𝑒−7

−7.71𝑒−7 ]

The performance index of the following form is constructed for LQT controller [23]:

𝐽 =1

2[𝐶𝑐𝑥(𝑘𝑓) − 𝑧(𝑘𝑓)]

𝑇 𝑆 [𝐶𝑐𝑥(𝑘𝑓) − 𝑧(𝑘𝑓)]

+1

2∑ [𝐶𝑐𝑥(𝑘) − 𝑧(𝑘)]𝑇𝑄 [𝐶𝑐𝑥(𝑘) − 𝑧(𝑘)] + 𝑢𝑇(𝑘)𝑅𝑢(𝑘)

𝑘𝑓−1

𝑘=𝑘0

(13)

with boundary condition 𝑥(𝑘0)= 𝑥0. Here 𝑧 is an 𝑛 dimensional reference vector, 𝑄 & 𝑆 are 𝑚 × 𝑚 symmetric, positive semidefinite matrices, 𝑅 is the actuator weighing matrix and 𝑘𝑓 represents final

location of the vector. Optimal control that optimizes the performance index is given by:

𝑢∗(𝑘) = −𝐿(𝑘)𝑥∗(𝑘) + 𝐿𝑔(𝑘)𝑔(𝑘 + 1) (14)

Quantities with an asterisk in superscript represent optimal quantities. Control gains 𝐿(𝑘) and 𝐿𝑔(𝑘) are

given by:

𝐿(𝑘) = [𝑅 + 𝐺𝑇𝑃(𝑘 + 1)𝐺]−1𝐺𝑇𝑃(𝑘 + 1)𝑆

𝐿𝑔(𝑘) = [𝑅 + 𝐺𝑇𝑃(𝑘 + 1)𝐺]−1𝐺𝑇 (15)

the vector 𝑔(𝑘) is given as:

𝑔(𝑘) = 𝑆𝑇{𝐼 − [𝑃−1(𝑘 + 1) + 𝐸]−1𝐸}𝑔(𝑘 + 1) + 𝑊𝑧(𝑘) (16)

with 𝑔(𝑘𝑓) = 𝑆𝑇𝑆 𝑧(𝑘𝑓) and W = 𝐶𝑐𝑇Q. Solution of the following matrix difference Riccati equation gives

matrix 𝑃(𝑘):

𝑃(𝑘) = 𝑆𝑇𝑃(𝑘 + 1)[𝐼 + 𝐸𝑃(𝑘 + 1)]−1𝑆 + 𝑉 (17)

with P(kf) = CTSC, V = CTQC and E = GR−1GT. In LQT actual output x∗(k) has to track desired input references z(k) using optimal controller. Controller feedback gain L(k) and feed-forward gain Lg(k) are

calculated offline by solving the Riccati equation. As can be seen, the control effort u∗(k) required to keep the output of plant close to desired input is obtained by minimizing the error between actual trajectory and reference trajectory. Substitution of optimal control law in equation of state gives optimal state x∗ as:

𝑥∗(𝑘 + 1) = [𝑆 − 𝐺𝐿(𝑘)]𝑥(𝑘) + 𝐺𝐿𝑔(𝑘)𝑔(𝑘 + 1) (18)

4. Simulations After deriving the finite element model of the structure, LQT controller is designed as per relations given in section 3. The technique is applied on a plate structure when it is disturbed and results are discussed in this section. First of all, zero references are given to all the states of the state-space model while applying LQT control law. Properties of piezoelectric patch and plate are tabulated in the table (1).

Table 1. Properties of piezoelectric patches and plate

Length Width Thickness 𝑑31 𝑑32 Density Young’s Relative (mm) (mm) (mm) (m/volt) (m/volt) (Kg/m3) modulus dielectric

(N/m2) constant

________________________________________________________________________________________

Piezoelectric patch 20 20 1 275E-12 275E-12 7500 4.8E10 3250

Plate 160 160 0.51 - - 7800 2.07E11 -

Weighting matrix 𝑆 and 𝑄 of controller using trial and error method are taken as:

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S =

[ 1 0 0 0 0 00 1 0 0 0 00000

0000

1 0 0 00 1 0 00 0 1 00 0 0 1]

, Q =

[ 1𝑒13 0 0 0 0 00 1𝑒17 0 0 0 00000

0000

1𝑒9 0 0 00 0 0 00 0 0 00 0 0 0]

(19)

In this simulation, the plate, theoriticaly is disturbed by vertically displacing the edge of plate opposite to the cantilevered edge of plate by 2 mm and then the LQT controller is applied on the smart plate system. All the three modes of vibration are given mean position i.e. zero references for tracking. Figure (2) presents time response of uncontrolled as well as controlled signals of the first three modes. It can be observed that all the modes are getting suppressed and approaching the mean position.

Fig. 2 Time response of uncontrolled, reference signal and LQT controlled vibration mode

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Next, reference signals for the first three modes of vibration are obtained by multiplying time responses of uncontrolled modal displacements shown in figure (2) by a factor of 0.5. Figure (3) presents performance of LQT controller when these new time-varying references are taken. It can be clearly observed in figure (3) that all the three modes of vibration have successfully and impressively tracked their respective time-varying reference signals.

Fig. 3 Time response of uncontrolled, reference signal and LQT controlled vibration mode

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The corresponding control voltages applied on the actuators are shown in figure (4). It can be confidently concluded that usage of LQT control in AVC application, allows to precisely dictating transient decay curves of individual modes simultaneously.

5. Conclusions Many control techniques have been applied on AVC so far but still scientis work on new techniques to get better performance in vibration suppression. In this paper for very first time the LQT controller has been used for AVC application. This technique has following advantages:

Full state measurement of system is not required

It can be applied on both modal displacement and actual vibration displacement

Desired multiple transient responses can be simultaneously achieved.

In this work, a novel technique has been used to control vibrations of a cantilevered plate structure. First, finite element model of the plate structure is derived and then it is converted into a state-space model. An optimal tracking control is then employed to track zero references. Using this technique one can successfully suppress multiple modes of vibration simultaneously in a structure using a simple procedure. Active vibration control is also successfully performed when non-zero transient decay curves are taken as references while applying LQT control. Present work presents a simple strategy to dictate transient response of individual modes of vibration while suppressing structural vibrations of an active structure.

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Fig. 4 Time response of control voltages applied on actuators

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