Du C., Xie L. Modeling and Control of Vibration in Mechanical Systems

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84 Modeling and Control of Vibration in Mechanical Systems

cessor based system consists of analog-to-dig ital converters to process sensor inputs

and digi tal-to-analog converters to convert the microprocessor’s output command

into an input signal to the actuator. The control logic, called the con trol algorithm,

programmed in t he computer uses sensor measurement to decide how much force

the actuator should apply.

4.4.1 Actuators

Piezoelectr ic materials can convert electrical current into motion, and vice versa.

They change shape when an electric curr ent passes through them, and th ey generate

an electri c signal when they ﬂex. Thus they can be used as actuators to create force

or motion, and as sensors to sense motion.

The materials used for high-precision actuation include electrostr ictive and mag-

netostrictive materials, which are similar to piezoelectr ic materials. These are ferro-

magnetic materials that expand or contract when subjected to an electric or a mag-

netic ﬁeld.

The previousl y mentio ned voice coil motor (VCM) actuator i n hard disk drives is

a linear actuator moving in one direction. Because of its similarity to a loudspeaker,

it is referred to as a voice coil motor. A V CM actuator has a coil of wire rigidly

attached to the structure and suspended in a permanent magnetic ﬁeld. It is driven

when a force is produ ced to accelerate it radiall y as a current is passed through the

coil.

Some actuators cannot provide enough force for larger applications such as vehi-

cle suspension and control of building motions. In building application, hydraulic

cylinders are usually used. With a working hydraulic pressure of commonly 2000

psi, a cylinder containing a piston whose area is only 1 square inch will generate 1

ton of force. Active vehicle susp ensions use hydraulic devices, electric motors, and

magneto-rheological ﬂuid dampers.

4.4.2 Active systems

An active system is appl ied to decrease vibrations by introduci ng dynamic loads

locally in the structure [8]. The dynamic loads are controlled by a processor in order

to minimize the vibratory level. The technology of generating loads is fundamental

in control strategy.

Active suspension

Several active principles are used for suspension s to isolate one structure from an-

other. One of them is internal load control to modify the distribution o f internal loads

in the structure. Its principle is to inject a set of dynamic loads into the structure in

order to minimize its vi bratory response. The loads depend on the vibrator y condi-

tion of the structure, whi ch is sensed with the help of a set of accelerometers or st rain

gauges. Hydraulic or electro-dynamic actuators suitably located on the structure are

used to inject the loads. The role of the actuators is to modify the distributio n of

vibratory energy for different modes to minimize the structure vibration s, instead of

dispersing the vibratory energy of the structure.

Classical Vibration Control 85

Active resonator

An active resonator wo rks by di splacing a mass wi th an actuator. It uses the dy-

namic ampliﬁcation of the mass to generate high loads using minimal energy. Differ-

ent types of actuator technologies such as hydraulic, electromagnetic, or piezoelec-

tric actuators, have been used to match the ﬁelds of application in terms of forces and

frequencies. We take an electromagnetic actuator as an example. The type of single

stage resonato r uses the principle of single stage mechanical syst ems wit h disp lace-

ment operated by an electromagnetic force. The motion equation of t he mass M of

a permanent magnet for small motion around its static posi tion is given by

M ¨x

+ c

˙x

+ k

= F

(t). (4.13)

With F

(t) = F

cos(ωt), the Laplace transform of the displ acement produced by

the mass is

(s) =

+ c

s + k

, (4.14 )

where k

and c

are respectively the stiffness and damping of the mass.

The load transmitted to the structure equals to

(t) = c

˙x

− k

−F

(t). (4.15 )

The mechanical parameters such as mass and stiffness are designed so that the

magnetic loads remain small and compatible with an acceptable energy consump-

tion. The natural frequency of the mass-spring system must be design ed as close as

possible to the frequency of t he vibration that needs to be controll ed.

A two-stage electromagnetic r esonator uses two masses called stages [8]. The

control force F

is introduced between the two masses by an electromagnetic load.

It is produced from a voltage V that is f ed to a power ampliﬁer to generate a magnetic

force via the current in the coil. The role of the ampliﬁer is to ensure the law between

the generated force F

and the control voltage V is l inear. The function of control

consists of tuning the system parameters such as mass and stiffness so that at the

control frequency, the force generated on the structure is ampliﬁed with respect to

the force F

, while maintain ing a small relative di splacement of the two stages.

The motion equations of the two masses are written as

¨x

+ c

( ˙x

− ˙x

) + k

−x

) = F

(t), (4.16)

¨x

+ c

( ˙x

− ˙x

) + c

˙x

+ k

−x

) + k

= −F

(t), (4.17)

with the primary stage M

and the secondary stage M

The load transmitted to the structure is

= c

˙x

+ k

. (4.18 )

86 Modeling and Control of Vibration in Mechanical Systems

4.4.3 Control strategy

When the syst em response is not acceptable, the value of the response is used to

generate additional forces according to rules or laws such that the modiﬁed response

behaves according to the design and within certain bounds. This results in a closed-

loop system that incorporates feedback contr ol, where the response is evaluated by a

sensor and is fed back to an actuator that generates a force or motion. The purpose of

designi ng a system with feedback force is to minimize unwanted behaviors and elim-

inate the effect of vibr ation on system performance in a desired manner. Feedback

control provides a mechanism for tailoring system behavior to speciﬁc standards and

needs. The block diagram depicting feedback control is i n Figure 4.14 . One can see

that the signal from a block goes concurrently to other blocks or a su mming point.

The summin g point indicates that the input and output signals are compared. The

difference is an error that generates a contr ol action. The disturbance or forcing is

also shown being input directly to the plant or being injected to the output of the

plant.

FIGURE 4.14

Generic feedback control system.

In order to properly design a feedback control system, performance must be de-

ﬁned in terms of system speciﬁcations. Standard performance measures are usually

deﬁned in terms of the step response. The general design objectives are the speed of

response, stabi lity, and accuracy or allowable error. The ﬁrst objective implies t hat it

is desirable for a control system to respond to a reference command rapidly. Stability

is the prerequisite of any active control syst em design. The allowable error represents

how close to the desired response the control force must bring the structure and how

much the control system can reject vibrations.

The controller produces control action or signal by using the comparison signal

of plant output with the desired reference value. There are a number of control

Classical Vibration Control 87

methods used by p eop le t o design control actions. Classical control is r epresented

by t he well-known proportional-integral-derivative (PID) control. Advanced control

techniques include robust and optimal controls, which will be intro duced brieﬂy in

the next chapter.

An adaptive control system i s one that can change or adapt either its gain values

or even its con trol algorithm to accommodate changing conditi ons. In mathematics

how the cont rol gains or the control algorithm should adapt to changing conditions

is difﬁcult. There are thus fewer practical applications of adaptive controls.

4.5 Conclusion

Various classical techniques of vibration control were discussed in this chapter and

some ty pical techniques such as using isolators, absorbers, a resonator, and suspen-

sion were introduced individually. Speciﬁcally, disk vibr ation reduction via stacked

disks as an application example was presented with detailed discussion and evalu-

ation. Subsequently, active vibrati on control was introduced with actuators, active

systems, and control strategies.

Introduction to Optim al and Robust Control

5.1 Introduction

This chapter intends to brieﬂy review the optimal and robust control. H

and H

∞

norms and their calculati on are ﬁrst introduced since t hey are two commonly used

measures of performance speciﬁcations of a control system. What follow are the

and H

∞

control problems and controller design via a linear matrix inequality

approach. Robust control of systems with multiplicative uncertainty and additive u n-

certainty, and the parametrization of all stable controllers, i.e., the so-called Youla

parametrization, will be investigated. Moreover, performance limitation of a feed-

back control system is discussed, which is necessary to help understand the vibration

control schemes in the later chapt ers.

5.2 H

and H

∞

norms

5.2.1 H

norm

The H

norm of a matrix transfer function G(s) analytic in Re(s) > 0 (open right-

half plane) is deﬁned as

kGk

sup

σ>0

{

2π

+∞

−∞

T race[G

∗

(σ + jω)G(σ + jω)]dω}, (5.1)

or equivalently [2]

kGk

2π

+∞

−∞

T race[G

∗

(jω)G(jω)]dω. (5.2)

Although kGk

can be computed fro m its deﬁnition, there are some simple alter-

natives taking advant age of a state-space representation of G (s).

LEMMA 5.1

[6] Consider a system G (s) with a state-space representation (A, B, C, D). If

90 Modeling and Control of Vibration in Mechanical Systems

A is stable and D = 0, then we have

kGk

= T race(B

B) = T race(CX

) (5.3)

where X

and Y

are the controllability and observability Gramians that can

be obtained from the following Lyapunov equations:

+ X

+ BB

= 0, (5.4)

+ Y

A + C

C = 0. (5.5)

We also consider a discrete-time linear time-invariant system G(z) with the fol-

lowing state-space representation:

x(k + 1) = Ax(k) + Bw(k), (5.6)

z(k) = Cx(k) + Dw(k), (5.7)

where x ∈ R

is the state, z ∈ R

is the controlled outpu t, w ∈ R

is the

disturbance input. Let T

denote the transfer function from th e input w to the

output z. Then the H

norm is deﬁned as

2π

T race[

−π

∗

jω

)dω]. (5.8)

By Parseval’s theorem, kT

can equivalent ly be deﬁned as

T race[

∞

k=0

g(k)g

(k)], (5.9)

where g(k) is the impulse response of T

Let the input w to the system be a w ide-band stationary stochastic process. The

norm of T

can also be interpreted as the RMS value of the output z(k) when

the system is subject to a white n oise having zero mean and unit variance. That is

E[z

z]. (5.10 )

The H

norm for the discrete-time system T

can be computed as

T race(D

D + B

B) =

T race(DD

+ CX

), (5.11)

where Y

and X

are the reachability and observability Gr amians that can be ob-

tained from the following Lyapunov equations:

−Y

+ BB

= 0 , (5.12)

A − X

+ C

C = 0. (5.13)

Introduction to Optimal an d Robust Control 91

The fo llowing theorem presents an alternative LMI condition for bounding the H

norm of the discrete-time system T

LEMMA 5.2

Consider a discrete-time t ransfer function T

of realization (A, B, C, D).

Given a scalar µ > 0, kT

< µ if and only if there exist X

= X

and

= Y

such that T race(Π) < µ and





Π CX

0 I





> 0, (5.14 )





0 I





> 0. (5.15)

Observe that (5.14) and (5.15) are linear in X

and Y

, and hence can be solved

by employing the LMI Tool [69] in MATLAB. The H

norm of the system can

be computed by minimizing µ using the function mincx in MATLAB Opt imization

Toolbox.

5.2.2 H

∞

norm

The H

∞

norm of a matrix transfer fu nction G(s) that is analytic and bounded in the

open ri ght-half plane is deﬁned as [3]

kGk

∞

:= sup

Re(s)>0

¯σ[G(s)] = sup

ω∈R

¯σ[G(jω)] = sup

w6=0

kzk

kwk

, (5.16)

where w and z are respectively the input and outpu t of G(s). A control engineering

interpretation of the inﬁnity n orm of a scalar transfer function G(s) is the distance in

the complex plane from the origin to the farthest point on the Nyquist plot of G, and

it also appears as the peak value on t he magnitude plot of |G(jω)|. Hence the H

∞

norm of a t ransfer function can, in principle, be obtained graphically.

In general, the H

∞

norm of a stable matrix transfer function can be read directly

from its singular value plots.

The H

∞

norm can also be computed in state-space.

LEMMA 5.3

[6] Let γ > 0 and G(s) : (A, B, C, D ) w ith A stable. Then kG(s)k

∞

< γ if

and only if ¯σ(D) < γ and the Hamiltonian matrix H has no eigenvalues on

the i m aginary axis, where

H :=



A + BR

−1

C BR

−1

−C

(I + DR

−1

)C − (A + BR

−1



, (5.17)

92 Modeling and Control of Vibration in Mechanical Systems

and R = γ

I − D

The following so-called Bounded Real Lemma gives a matrix inequality condition

for the system G(s) to have a pre-speciﬁed level of H

∞

norm.

LEMMA 5.4

Continuous-time Bounded Real Lemma Consider a continuous-time transfer

function G(s) of realization G(s) = C(sI − A)

−1

B + D, where A is a stable

matrix. Given a scalar λ > 0, the H

∞

norm k Gk

∞

< λ if and only if there

exists X = X

> 0 such that





X + XA XB C

X − λI D

C D − λI





< 0 . (5.18 )

In the discrete-time case, the H

∞

norm of a stable transfer function matrix T

(z)

with realization (A, B, C, D) is deﬁned as

∞

= sup

ω∈[0,2π]

¯σ[T

jω

)] = sup

w6=0

kzk

kwk

, (5.19 )

where w and z are respectively the input and output of T

The following discrete-time Bounded Real Lemma provid es a l inear matri x in-

equality conditio n for the system T

to have kT

∞

< γ.

LEMMA 5.5

Discrete-time Bounded Real Lemma Consider a discrete-time and a stable

transfer function matrix T

(z) with a state-space realization (A, B, C, D).

Then, for some given λ > 0, kT

∞

< λ if and only if there exists X =

> 0 such that





XA −X A

XB C

XA B

XB − λ

I D

C D −I





< 0. (5.20 )

5.3 H

optimal control

5.3.1 Continuous-time case

We consider the closed-loop system described by the block d iagram in Figure 5.1.

The cont inuous-time linear time-invariant plant P (s) is descri bed by the state-space

equations:

˙x(t) = Ax(t) + B

w(t) + B

u(t), (5.21)

Introduction to Optimal an d Robust Control 93

z(t) = C

x(t) + D

w(t) + D

u(t), (5.22 )

y(t) = C

x(t) + D

w(t) + D

u(t), (5.23 )

where x ∈ R

is the state, y ∈ R

is the measurement output, z ∈ R

is the

controlled output, w ∈ R

is the disturbance input, u ∈ R

is the control input ,

and A, B

, B

, C

, D

, C

, and D

are of approp riate dimensions. We

assume D

= 0 without loss of generality [89].

Intr oduce the following dynamic output feedback controller C(s):

˙x

(t) = A

(t) + B

y(t), (5.24)

u(t) = C

(t) + D

y(t). (5.25)

Denote ξ = [x

]

. From (5.21)−(5.23) and (5.24)−(5.25), the closed-loop

system is given by

ξ(t) =

Aξ(t) +

Bw(t), (5.26 )

z(t) =

Cξ(t) +

Dw(t), (5.27)

where

A =



A + B



B =



+ B



, (5.28)

C = [C

+ D

] ,

D = D

+ D

. (5.29)

The continuous-time H

control problem is to ﬁnd a pr oper, real rational controller

C(s) that stabilizes P internally and minimizes the H

norm of the closed-loop

transfer function matrix T

from w to z in (5.26)−(5.27).

Assume th at the system (5.21)−(5.23) satisﬁes the following conditions:

Assumption 5.1

(1). D

is of full column rank;

(2). The subsystem (A, B

, C

, D

) has no i nvariant zeros o n th e imaginary axis;

(3). D

is of full row rank ;

(4). The subsystem (A, B

, C

, D

) has no invariant zeros on the i maginary

axis.

Let X

≥ 0 and Y

≥ 0 be the solutions of the following Riccati equatio ns:

+ X

A −(X

+ C

)(D

)

−1

+ C

)

= 0, (5.30 )

+ AY

− (Y

+ B

)(D

)

−1

+ B

)

= 0. (5.31 )

According to th e H

optimal con trol theory, an H

optimal con troller can be ob-

tained as [7]

= A + B

F + KC

, B

= −K, C

= F, D

= 0 , (5.32)