32.9 Active Control of Vibration

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Passive control of vibration is relatively simple and straightforward. Although it is robust, reliable, and

economical, it has its limitations. Note that the control force that is generated in a passive device depends

entirely on the natural dynamics. Once the device is designed (i.e., after the parameter values for mass,

damping constant, stiffness, location, etc. are chosen), it is not possible to adjust the control forces that

are naturally generated in real time. Furthermore, in a passive device there is no supply of power from an

external source. Hence, even the magnitude of the control forces cannot be changed from their natural

values. Since a passive device senses the response of the system as an integral process of the overall

dynamics of the system, it is not always possible to directly target the control action at particular

responses (e.g., particular modes). This can result in incomplete control, particularly in complex and

high-order (e.g., distributed-parameter) systems. These shortcomings of passive control can be overcome

using active control. Here, the system responses are directly sensed using sensor-transducer devices, and

control actions of specific desired values are applied to desired locations/modes of the system.

32.9.1 Active Control System

Figure 32.32 presents a schematic diagram of an active control system. The mechanical dynamic system

whose vibrations need to be controlled is the plant or process. The controller is the device that generates

the signal (or command) according to some scheme (or control law) and controls vibrations of the plant.

The plant and the controller are the two essential components of a control system. Usually, the plant must

Sensor/

Transducer

Power

(For Active Sensors)

Vibrating System

(Plant, Process)

Response

Disturbance

Excitation

Actuator

Power

Drive

Excitation

Signal

Conditioning

Power

Controller

(Digital or

Analog)

Power

Control

Signal

Reference

Command

Signal

Conditioning

Power

Feedback

Signal

FIGURE 32.32 A system for active control of vibration.

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© 2005 by Taylor & Francis Group, LLC

be monitored and its response must be measured using sensors providing feedback into the controller.

Then, the controller compares the sensed signal with a desired response specified externally, and uses

the error to generate a proper control signal. In this manner, we have a feedback control system. In the

absence of a sensor and feedback, we have an open-loop control system. In feed-forward control, the

excitation (i.e., input signal), not the response (i.e., output signal), is measured and used (i.e., fed

forward into the controller) for generating the control signal. Both feedback and feedforward schemes

may be used in the same control system.

The actuator that receives a control signal and drives the plant may be an integral part of the plant

(e.g., the motor that drives the blade of a saw). Alternatively, it may have to be added specifically as an

external component for the control actuation (e.g., a piezoelectric or electromagnetic actuator for

controlling blade vibrations of a saw). In the former case, in particular, proper signal conditioning is

needed to convert the control signal to a form that is compatible with the existing actuator. In the latter

case, both the controller and the actuator must be developed in parallel for integration into the plant. In

digital control, the controller is a digital processor. The control signal is in digital form and, typically, it

has to be converted into the analog form prior to using in the actuator. Hence, digital-to-analog

conversion (DAC) is a form of signal conditioning that is useful here. Furthermore, the analog signal that

is generated may have to be filtered and amplified to an appropriate level for use in the actuator. It follows

that filters and amplifiers are signal conditioning devices, which are useful in vibration control. In

software control, the control signal is generated by a computer, which functions as the digital controller. In

hardware control, the control signal is rapidly generated by digital hardware without using software

programs. Alternatively, analog control may be used where the control signal is generated directly using

analog circuitry. In this case, the controller is quite fast and it does not require DAC. Note that the

actuator may need high levels of power. Furthermore, the controller and associated signal conditioning

will require some power. The need for an external power source for control distinguishes active control

from passive control.

In a feedback control system, sensors are used to measure the plant response, which enables the

controller to determine whether the plant operates properly. A sensor unit that “senses” the response may

automatically convert (transduce) this “measurement” into a suitable form. A piezoelectric accelerometer

senses acceleration and converts it into an electric charge, an electromagnetic tachometer senses velocity

and converts it into a voltage, and a shaft encoder senses a rotation and converts it into a sequence of

voltage pulses. Hence, the terms sensor and transducer are used interchangeably to denote a sensortransducer

unit. The signal that is generated in this manner may need conditioning before feeding into

the controller. For example, the charge signal from a piezoelectric accelerometer has to be converted to a

voltage signal of appropriate level using a charge amplifier, and then it has to be digitized using an

analog-to-digital converter (ADC) for use in a digital controller. Furthermore, filtering may be needed to

remove measurement noise. Hence, signal conditioning is usually needed between the sensor and the

controller as well as between the controller and the actuator. External power is required to operate active

sensors (e.g., potentiometer) whereas passive sensors (e.g., electromagnetic tachometer) employ selfgeneration

and do not need an external power source. External power may be needed for conditioning

the sensor signals. Finally, as indicated in Figure 32.32, a vibrating system may have unknown

disturbance excitations, which can make the control problem particularly difficult. Removing such

excitations at the source level through proper design or vibration isolation is desirable, as discussed

above. However, in the context of control, if these disturbances can be measured or some information

about them is available, then they can be compensated for within the controller itself. This is, in fact, the

approach of feedforward control.

32.9.2 Control Techniques

The purpose of a vibration controller is to excite (activate) a vibrating system in order to control its

vibration response in a desired manner. In the present context of active feedback control, the controller

uses measured response signals and compares them with their desired values in its task of determining an

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© 2005 by Taylor & Francis Group, LLC

appropriate action. The relationship that generates the control action from a measured response (and a

desired value for the response) is called a control law. Sometimes, a compensator (analog or digital,

hardware or software) is employed to improve the system performance or to enhance the controller so

that the task of control is easier. However, for our purpose, we may consider a compensator as an integral

part of the controller and thus a distinction between the two is not made.

Various control laws, both linear and nonlinear, have been developed for practical applications. Many

of them are suitable in vibration control. A comprehensive presentation of all such control laws is outside

the scope of this book. We will give several linear control laws that are common and representative of

what is available. These techniques are based on a linear representation (linear model) of the vibrating

system (plant). Even when the overall operating range of a plant (e.g., robotic manipulator) is nonlinear,

it is often possible to linearize the vibration response (e.g., link vibrations and joint vibrations of a robot)

about a reference configuration (e.g., robot trajectory). These linear control techniques would be still

suitable even though the overall dynamics of the system is nonlinear.

32.9.2.1 State-Space Models

In applying many types of control techniques, it is convenient to represent the vibrating system (plant) by

a state-space model. This is simply a set of ordinary first-order differential equations, which could be

coupled or nonlinear, and could have time-varying parameters (time-variant models). Here, we limit our

discussion to linear and time-invariant state-space models. Such a model is expressed as

x_ ¼ Ax þ Bu ð32:131Þ

y ¼ Cx þ Du ð32:132Þ where

x ¼ ½x1; x2; …; xn􀀉T ¼ state vector (nth order column)

u ¼ ½u1; u2; …; ur 􀀉T ¼ input vector (rth order column)

y ¼ ½y1; y2; …; ym􀀉T ¼ output vector (mth order column)

A ¼ system matrix (n £ n square)

B ¼ input gain matrix (n £ r)

C ¼ measurement gain matrix (m £ n)

D ¼ feedforward gain matrix (m £ r)

Usually, for vibrating systems, it is possible to make D ¼ 0; and hence we will drop this matrix in the

sequel. Furthermore, although a state variable xi need not have a direct physical meaning, an output

variable yj should have some physical meaning and, in typical situations, should be measurable as well.

The input variables are the “control variables” and are used for controlling the system (plant). The output

variables are the “controlled variables,” which correspond to the system response and are measured for

feedback control.

It can be verified that the eigenvalues of the system matrix A occur in complex conjugates of the form

2zivi ^ j

ffiffiffiffiffiffiffiffi

1 2 z2i

q

vi in the damped oscillatory case, or as ^jvi in the undamped case, where vi is the ith

natural frequency of the system and zi is the corresponding damping ratio (of the ith mode). The

mathematical verification requires some linear algebra. An intuitive verification can be made since

Equation 32.131 is an equivalent model for a system having the traditional mass – spring – damper model

My€ þ Cy_ þ Ky ¼ f ðtÞ ð32:133Þ

where M ¼ mass matrix, C ¼ damping matrix, K ¼ stiffness matrix, f(t) ¼ forcing input vector, and

y ¼ displacement response vector. Where both models (Equation 32.131 and Equation 32.133), are

equivalent they should have the same characteristic equation, which by its roots determines the natural

frequencies and modal damping ratios. This is the case because we are simply looking at two different

mathematical representations of the same system. Hence, the parameters of its dynamics, such as vi and

zi; should remain unchanged. In fact, the state-space mode (Equation 32.131) is not unique, and different

versions of state vectors and corresponding models are possible. Of course, all of them should have the

Vibration Design and Control 32-63

© 2005 by Taylor & Francis Group, LLC

same characteristics polynomial (and hence, the same vi and ziÞ: One such state-space model may be

derived from Equation 32.133 as follows:

Define the state vector as

x ¼

y

y_

" #

and u ¼ fðtÞ ð32:134Þ

Since (for nonsingular M, as required), Equation 32.133 may be written as

y€ ¼ 2M21Ky 2 M21Cy_ þ M21f ðtÞ ð32:135Þ

we have

x_ ¼

0 I

2M21K 2M21C

" #

x þ

0

M21

" #

u ð32:136Þ

This is a state-space model that is equivalent to the conventional model (Equation 32.133), and can be

shown to have the same characteristic equation. The development of a state-space model for a vibrating

system can be illustrated using an example.

Example 32.10

Consider a machine mounted on a support

structure, modeled as in Figure 32.33. Using the

excitation forces f1ðtÞ and f2ðtÞ as the inputs and

the displacements y1 and y2 of the masses m1 and

m2 as the outputs, develop a state-space model for

this system.

Solution

Assume that the displacements are measured from

the static equilibrium positions of the masses.

Hence, the gravity forces do not enter into the

formulation. Newton’s Second law is applied to

the two masses; thus

m1y€1 ¼ f1 2 k1ðy1 2 y2Þ 2 b1ðy_1 2 y_2Þ

m2y€2 ¼f22k1ðy22y1Þ2b1ðy_22y_1Þ2k2y22b2y_2

The following state variables are defined:

x1 ¼ y1; x2 ¼ y_1; x3 ¼ y2; x4 ¼ y_2

Also, the input vector is u ¼ ½u1 u2􀀉T and the output vector is y ¼ ½y1 y2􀀉T: Then, we have

x_1 ¼ x2

m1x_2 ¼ u1 2 k1ðx1 2 x3Þ 2 b1ðx2 2 x4Þ

x_3 ¼ x4

m2x_4 ¼ u2 2 k1ðx3 2 x1Þ 2 b1ðx4 2 x2Þ 2 k2x3 2 b2x4

f2(t)

k1

k2

Support

Structure

y1

y2

m1 Machine

m2

b1

f1(t)

b2

FIGURE 32.33 A model of a machine mounted on

support structure.

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© 2005 by Taylor & Francis Group, LLC

Accordingly, the state-space model is given by Equation 32.131 and Equation 32.132 with

A ¼

0 1 0 0

2k1=m1 2b1=m1 k1=m1 b1=m1

0 0 0 1

k1=m2 b1=m2 ðk1 þ k2Þ=m2 ðb1 þ b2Þ=m2

2

66666664

3

77777775

; B ¼

0 0

1=m1 0

0 0

0 1=m2

2

66666664

3

77777775

;

C ¼

1 0 0 0

0 0 1 0

2

4

3

5; and D ¼ 0

Also, note that the system can be expressed as

m1 0

0 m2

" #

y€ þ

b1 2b1

2b1 ðb1 þ b2Þ

" #

y_ þ

k1 2k1

2k1 ðk1 þ k2Þ

" #

y ¼ f ðtÞ

Its characteristic equation may be expressed as the determinant equation:

det

m1s2 þ b1s þ k 2b1s 2 k1

2b1s 2 k1 m2s2 þ ðb1 þ b2Þs þ ðk1 þ k2Þ

" #

¼ 0

It can be verified through direct expansion of the determinants that this equation is equivalent to the

characteristic equation of the matrix A, as given by detðlI 2 AÞ ¼ 0; or

det

l 21 0 0

k1=m1 l þ b1=m1 2k1=m1 2b1=m1

0 0 l 21

k1=m2 2b1=m2 2ðk1 þ k2Þ=m2 l 2 ðb1 þ b2Þ=m2

2

6666664

3

7777775

¼ 0

Note that, in the present context, x and y represent the vibration response of the plant and the control

objective is to reduce these to zero. We will give some common control techniques that can achieve

this goal.

32.9.2.2 Position and Velocity Feedback

In this technique, the position and velocity of each DoF is measured and fed into the system with sign

reversal (negative feedback) and amplification by a constant gain. Because velocity is the derivative of

position and since the gains are constant (i.e., proportional), this method falls into the general category

of proportional-plus-derivative (PD or PPD) control. In this approach, it is tacitly assumed that the

degrees of freedom are uncoupled. Then, control gains are chosen so that the DoF in the controlled

system are nearly uncoupled, thereby justifying the original assumption. To explain this control method,

suppose that a DoF of a vibrating system is represented by

my€ þ by_ þ ky ¼ uðtÞ ð32:137Þ

where y is the displacement (position) of the DoF and u is the excitation input that is applied. Now

suppose that u is generated according to the (active) control law

u ¼ 2kcy 2 bcy_ þ ur ð32:138Þ

where kc is the position feedback gain and bc is the velocity feedback gain. The implication here is that the

position y and the velocity y_ aremeasured and fed into the controller which in turn generates u according

to Equation 32.138. Also, ur is some reference input that is provided externally to the controller.

Vibration Design and Control 32-65

© 2005 by Taylor & Francis Group, LLC

Then, substituting Equation 32.138 into Equation 32.137, we obtain

my€ þ ðb þ bcÞy_ þ ðk þ kcÞy ¼ ur ð32:139Þ

The closed-loop system (the controlled system) now behaves according to Equation 32.139. The control

gains bc and kc can be chosen arbitrarily (subject to the limitations of the physical controller, signal

conditioning circuitry, the actuator, etc.) and may even be negative. In particular, by increasing bc, the

damping of the system can be increased. Similarly, by increasing kc the stiffness (and the natural

frequency) of the system can be increased. Even though a passive spring and damper with stiffness kc and

damping constant bc can accomplish the same task, once the devices are chosen it is not possible to

conveniently change their parameters. Furthermore, it will not be possible to make kc or bc negative in

this case of passive physical devices. The method of PPD control is simple and straightforward, but the

assumptions of linear uncoupled DoF place a limitation on its general use.

32.9.2.3 Linear Quadratic Regulator Control

This is an optimal control technique. Consider a vibrating system that is represented by the linear statespace

model:

x_ ¼ Ax þ Bu ð32:131Þ

Assume that all the states x are measurable and all the system modes are controllable. Then, we use the

constant-gain feedback control law:

u ¼ Kx ð32:140Þ

The choice of parameter values for the feedback gain matrix K is infinite. Therefore, we can use this

freedom to minimize the cost function:

J ¼

1

2

ð1

t ½xTQx þ uTRu􀀉dt ð32:141Þ

This is the time integral of a quadratic function in both state and input variables, and the optimization

goal may be interpreted as bringing x down to zero (regulating x to 0), but without spending a rather

high control effort. Hence, the name linear quadratic regulation (LQR). In addition, Q and R are

weighting matrices, with the former being at least positive semidefinite and the latter positive definite.

Typically, Q and R chosen as diagonal matrices with positive diagonal elements whose magnitudes are

determined by the degree of relative emphasis that should be given to various elements of x and u. It is

well known that K that minimizes the cost function (Equation 32.141) is given by

K ¼ 2R21BTKr ð32:142Þ

where Kr is the positive-definite solution of the matrix Riccati algebraic equation

KrA þ ATKr 2 KrBR21BTKr þ Q ¼ 0 ð32:143Þ

It is also known that the resulting closed-loop control system is stable. Furthermore, the minimum

(optimal) value of the cost function (Equation 32.141) is given by

Jm ¼

1

2

xTKrx ð32:144Þ

where x is the present value of the state vector. Major computational burden of the LQR method is in the

solution (Equation 32.143). Other limitations of the technique arise due to the need to measure all the

state variables (which may be relaxed to some extent). Although stability of the controlled system is

guaranteed, the level of stability that is achieved (i.e., stability margin or the level of modal damping)

cannot be directly specified. Further, robustness of the control system in the presence of model errors,

unknown disturbances and so on, may be questionable. Besides, the cost function incorporates an

integral over an infinite time duration, which does not typically reflect the practical requirement of rapid

vibration control.

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32.9.2.4 Modal Control

The LQR control technique has the serious limitation of not being able to directly achieve specified levels

of modal damping, which may be an important goal in vibration control. The method of modal control

that accomplishes this objective is pole placement, where poles (eignevalues) of the controlled system are

placed at specified values. Specifically, consider the plant (Equation 32.131) and the feedback control law

(Equation 32.140). Then, the closed-loop system is given by

x_ ¼ ðA þ BKÞx ð32:145Þ

It is well known that if the plant (A, B) is controllable, then a control gain matrix K can be chosen that

will arbitrarily place the eigenvalues of the closed-loop system matrix A þ BK. Based on the given

assumptions, the modal control technique assigns not only the modal damping but also the damped

natural frequencies at specified values. The assumptions given above are quite stringent but they can be

relaxed to some degree. However, a shortcoming of this method is the fact that it does not place a

restriction on the control effort, for example, as the LQR technique does, in achieving a specified level of

modal control.

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