1997 - A Comparison of Semi-Active Control Strategies for the MR Damper

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A Comparison of Semi-Active Control Strategies for the MR Damper

S.J. Dyke' and

B.F.

Spencer Jr.

'Dept.

of

Civil Engineering, Washington Univ., St.

Louis, M O

63130,

U.S.A.

2Dept. of Civil Enginee ring and Ge ol. Sci., Univ. of Notre Dame, Notre Dame, IN 46556,

U.S.A.

Abstract

Because the behavior

of

semi-active de vices is often highly

nonlineal; one of the ma in challenges in the application of

this technology is the development of appropriate control

algorilhms. Herein, a number of recently proposed semi-

active control ulgorithms are evaluated for use wit?i the

magrietorheological M R ) damper; an innovative semi-ac-

tive control device th at appears to be particularly prom is-

ing fo r civil engineering app lications. T he perj5ormance of

the resulting control systems are compared through simu-

lation, and the advantages

of

each algorithm are dis-

cussed. The results demonstrate that the pelformanee of

the control system is highly depend ent on the choice of al-

gori thm employed.

1: Introduction

The magnetorheological (MR) damper is a semi-ac-

tive control device that is capable of generating the magni-

tude of forces necessary for full-scale applications, while

requiring only

a

battery for power [2,

3, 211.

Additionally,

this device is offers highly reliable operation at

a

modest

cost and its performance is relatively insensitive to temper-

ature fluctuations or impurities in the fluid. However, be-

cause of the inherent nonlinear nature of these devices,

one of the challenging aspects of utilizing this technology

to achieve high levels of performance is in the develop-

ment of appropriate control algorithm that can take advan-

tage of the unique characteristics of the device.

A variety of semi-active devices and control algo-

rithms have been proposed for seismic control

[7-14,

201.

Because the characteristics of the various semi-active de-

vices are different (e.g., variable friction, fluid-orificing,

controllable fluids, etc.) , a control algorithm that performs

well for one device

may

not be suitable for use with anoth-

er device. The focus of this paper is to investigate a num-

ber of recently proposed control strategies for use with the

MR dampcr. In simulation,

one

cxample

is

considered in

which these methods are applied to control a three-story

structure using an MR damper. To assess the effectiveness

of each of the control algorithms, they will be evaluated in

their ability to reduce the peak responses

for

the

N-S

com-

ponent of the El Centro earthquake excitation. The con-

trolled responses are compared to the uncontrolled and

passive responses.

2: Mechanical Model

of

the MR Damper

To

evaluate the potential of MR dampers in structural

control applications, the simple mechanical model depict-

ed in Fig.

1

was developed and shown to accurately predict

the behavior

of

an MR damper over a broad range of in-

puts [17, 181. The model is based on the response of a pro-

totype MR damper, obtained for evaluation from the Lord

Corporation. The equations governing the force

f

predict-

ed by this model are

(3)

where

z

is an evolutionary variable that accounts for the

history dependence of the response. The model parameters

depend on the voltage v to the current driver as follows

a a , + a b u , c1 = C l a + C l b U y O =

C O a + C O b U 4)

where U is given as the output of the first-order filter

li = q ( u - v ) . ( 5 )

Eq.

(5)

is necessary to model the dynamics involved in

reaching rheological equilibrium and in driving the elec-

tromagnet in the MR damper [17, 181.

3:

Control Algorithms

Consider

a

seismically excited structure controlled

with a single MR damper. Assuming that the forces pro-

vided by the MR damper are adequate to keep the re-

sponse of the primary structure from exiting the linear

region, then the equations of motion can be written as

M,X

C,X K,x = Af

-

M,rZg

6 )

where x is

a

vector of the relative displacements of the

floors of the structure,

x

is

a

one-dimensional ground

ac-

celeration, f is the measured control force, defined by

Eqs.

(1-.5), r is a column vector of ones, and A is a vector de-

0 8186 8218 3/97

10.00

997

IEEE

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Figure

1

Simple Mechanical R Damper Model

termined by the position of the

MR

damper in the struc-

ture. This equation can be written in state-space form as

Z

= A z + B f + E f 7)

y

=

C z + D f + v (8)

g

where z is the state vector,

y

is the vector of measured

outputs, and

v

is the measurement noise vector. For these

applications, the measurements typically available for con-

trol force determination include the absolute acceleration

of selected points on the structure, the displacement of the

control device, and a measurement of the control force.

A variety of approaches have been proposed in the lit-

erature for the control of semi-active devices. Subsequent-

ly, a selection of these approaches will be presented and

evaluated for a numerical example. In developing the con-

trol laws, note that i) the control voltage is restricted to the

range

v

E

[0,

V,,,] , and ii) for a fixed set of states, the

magnitude of the applied force

I f

increases when

v

in-

creases, and decreases when v decreases.

3.1:

Control Based on Lyapunov Stability Theory

In some cases it is possible to employ Lyapunov's di-

rect approach to stability analysis in the design of a feed-

back controller [l]. The approach requires the use of a

Lyapunov function, denoted V z) , which must be a posi-

tive definite function of the states of the system, z . Ac-

cording to Lyapunov stability theory, if the rate of change

of the Lyapunov function,

V z)

, s negative semi-definite,

the origin is stable i.s.L. (in the sense of Lyapunov). Thus,

in determining the control law, the goal is to choose a con-

trol input which will result in making

V

as negative as

possible. An infinite number of Lyapunov functions may

be chosen, which may result in a variety of control laws.

Leitmann

[141

applied Lyapunov's direct approach for

the design of a semi-active controller. In this approach, a

Lyapunov function is chosen of the form

(9)

1 2

V ( Z > =

~ I l Z l l P

where I

IzI p

is the P-norm of the states defined by

(10)

1 2

llzllp

=

[Z PZI

and P is a real, symmetric, positive definite matrix. In the

case of a linear system, to ensure

V

is negative definite,

the matrix P is found using the Lyapunov equation

A P + P A =

Qp (1

1)

for a positive definite matrix

Q p .

The derivative of the

Lyapunov function for a solution of Eq. (7)

is

V

= - z Q,z z PBf z PEXg .

(12)

Thus, the control law which will minimize V is

v

= V,,,H(-z PBf),

(13)

where H (

.)

is the Heavisicle step function. Notice that this

algorithm is classified as a bang-bang controller, and is de-

pendent on the sign of the measured control force and the

states of the system. To implement this algorithm, a Kal-

man filter is used to estimate the states based on the avail-

able measurements

i .e . ,

:MR damper displacement and

structural accelerations). Thus, in this algorithm, better

performance is expected when measurements of the re-

sponses of the full structure are used.

3.2:

Decentralized Bang-Bang Control

McClamroch and Gavin [161 used a similar approach

to develop the decentralized bang-bang control law. In this

approach, the Lyapunov function was chosen to represent

the total vibratory energy in the structure (kinetic plus po-

tential energy), as in

(14)

Using Eq.

(6) ,

he rate of change of the Lyapunov function

is then

V

=

;x K,x

: X

i

x,) M,(x +

rx,

V =

ix K,X

+ ( X

rig) (-,X -K,x Af) .

(15)

In this expression, the only way to directly effect V is

through the last term. To achieve the goal of making V a

large and negative (maximizing the rate at which energy is

dissipated), the following control law is chosen

(16)

Note that, because the only non-zero terms in the

A

vector

are those corresponding to the location of the MR damper,

this control law requires only measurements of the floor

velocities and applied forces. When the damper is located

in the upper floors, the relative velocity is needed. Interest-

ingly, when a semi-active device is located between the

ground and first floor, the absolute velocity of the first

floor is required, which is not readily available. Therefore,

to implement this control algorithm, one would approxi-

mate the absolute velocity (obtain the pseudo velocity) by

integrating the absolute acceleration, as in [191.

v

= V m a x H - X

T i J A f ) .

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3.3:

Clipped-Optimal Control

One algorithm that has been shown to be effective for

use with the MR damper is the clipped-optimal control ap-

proach, proposed by Dyke, et al. [7-91. The clipped-opti-

mal control approach is to design a linear optimal

controller K,( s) that calculates a desired control force

f

based on the measured structural responses y

and the

measured force f applied to the structure, i.e.,

f = L-l{ K , O L { }}

where L . } is the Laplace transform. Because the force

generated in the MR damper is dependent on the responses

of the structural system, the desired optimal control force

f cannot always be produced by the MR damper. Only

the control voltage v can be directly controlled. Thus, a

force feedback loop is incorporated to induce the MR

damper to generate approximately the desired optimal

control force

f , .

To this end, the command signal

v

is se-

lected according to the control law

v =

V , , x H ( { f , - f l f ) .

(18)

Although a variety of approaches may be used to design

the optimal controller, H , LQG methods are advocated

because of their successful application in previous studies

[5-111. The approach to optimal control design is dis-

cussed further in [8 ,9] .

3.4:

Modulated Homogeneous Friction

Another semi-active control algorithm was proposed

for use with a variable friction damper [13]. In this ap-

proach, at every occurrence of a local extrema in the defor-

mation of the device

(i.e.,

when the relative velocity

between the ends of the semi-active device is zero), the

normal force applied at the frictional interface is updated

to a new value. At each local minima or maxima in the de-

formation, the normal force,

N

is chosen to be propor-

tional to the absolute value of the deformation

of

the semi-

active device. The control law is written [131

N t )

=

glP[A(t)ll (19)

where g is a positive gain, and the operator P [ a] (referred

to as the prior-local-peak operator) is defined as

P[A(t)] = A t - s ) , where

s =

{min

x>O:A(t-x)=O},

(20)

defining

A ( t - )

as the most recent local extrema in the

deformation.

Because this algorithm was developed for use with a

variable friction device, the following modifications were

necessary to apply it to the MR damper: i) there is no need

to check if the force is greater than p N where

p

is the

coefficient of friction, because the MR damper is not sub-

ject to static friction, and ii) a force feedback loop was

used to induce the MR damper to produce approximately

the frictional force corresponding to the desired normal

force. Thus, the goal is to generate a desired control force

with a magnitude

where the proportionality constant

g,

has units of stiff-

ness. The resulting control law is

An appropriate choice

of

g will keep the force f ithin

the operating envelope of the MR damper a majority of the

time, allowing the MR damper force to closely approxi-

mate the desired force. Additionally, notice that this con-

trol law requires only measurements of applied force and

the relative displacements of the control device.

4: Numerical Example

The performance of the control algorithms presented

previously are now evaluated in one example through nu-

merical simulation. A model of a three-story building con-

figured with a single MR damper is considered. The MR

damper is rigidly connected between the ground and the

first floor, as shown in Fig. 2.The governing equations ca

be written in the form of Eq. (6) by defining

98.3

0

175 -50 0 N. ec

Ms = 0 98.3 0

k g ,

Cs = -50 100 -50

I

0 1 0

-50

5112.0 -6.84 0

N

0 -6.84 6.84

I:],

3

, =

10

-6.84 13.7 -6.84 ;

A

= 0 r = 1 .

This system is a simple model of the scaled, three-story,

test structure, described in [6, 91 which has been used in

previous active control studies at the Structural Dynamics

and Control / Earthquake Engineering Laboratory at the

University of Notre Dame. Because the MR damper is at-

tached between the first floor and the ground, its displace-

ment is equal to the first floor relative displacement, i.e.,

x

=

x1

in Eqs. (1-3).

In this example, the structural measurements available

for calculating the control action include the absolute ac-

celerations of the structure, and the MR damper displace-

ment i .e . , y =

[ f a l xa2 jin3 x,]).

Thus, the governing

equations can be written in the form of Eqs. (7-8) by de-

fining

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A = [

M,0

, -M, 1,C,

, B = [ MI

A , E = - /

r r

The MR damper parameters used in this study are

c O ,

= 8 Nseckm, C o b

=

6 Nsec/cmN, k ,

=

50 Nkm,

c l a =

290

Nseckm, C l b

=

5 NseclcmN, k , = 12 Nkm,

xo =

0

cm, a = 100, b = 450 v', y = 363 cm-2, = 363 cm-

', A = 301,

n

= 2 ,

q

= 190 sec-'. These parameters were

identified based on the prototype MR damper tested at the

University of Notre Dame

[9,

lo].

In simulation, the model of the structure is subjected

to the NS component of the 1940 El Centro earthquake.

The simulations were performed in MATLAB [15]. Be-

cause the system under consideration is a scaled model,

the earthquake was reproduced at five times the recorded

rate. The maximum structural responses of each system

are presented in Table 1. Here, xi is the displacement of

the i th floor relative to the ground, d j is the interstory

drift (i.e.,

x i

i ,

.fa

is the absolute acceleration of the

th floor, and f is the applied control force.

As

a

basis for comparison, two cases are considered in

which the MR damper is employed in

a

passive mode. In

the first case, designated

pussive-off,

the command voltage

to the MR damper is held at 0 V. The second passive case

the voltage to the MR damper is held at the maximum

voltage level (2.25 V) and is denoted as passive-on. From

the results, both passive systems are able to achieve

a

rea-

sonable level

of

performance. However, notice that the

passive-on system results in larger maximum accelerations

and interstory displacements than the passive-off system.

Apparently, choosing a passive device that produces the

largest damping forces may not be the most effective ap-

proach to protective system design.

A variety of

Q p

matrices were tested for the

Lyapunov-based controlle:r, and the best results were

achieved when the

Qp

matrix was chosen

as a

6 x 6 ma-

trix, with nonzero values only in the (l,l), (2,2), and (3,3)

positions. While using a lower control force, this control-

ler reduces the peak interstory displaccment and peak ab-

solute accclcration by an additional 6.9 and

19 ,

respectively, over the best passive case.

In this example, the decentralized bang-bang control-

ler appears to be quite effective at reducing the maximum

acceleration (33 below the better passive case). Howev-

er, this controller was not effective at reducing the relative

displacements or interstory displacements. The maximum

interstory displacement

is

slightly larger than that of the

best passive case and the peak force is quite large. Because

this control algorithm is attempts to minimize the total en-

ergy, it appears to be imitating

a

base isolation system, re-

sulting in

a

large relative displacement at the first floor.

The results in Table I show that the performance of

the system employing

a

clipped-optimal controller sur-

passes that of both passive systems considered. The

clipped-optimal controlleir reduces the maximum third

floor relative displacement and interstory displacement by

an additional 31 and 28 , respectively, as compared to

the best passive response. In addition, this controller re-

duces the maximum floor acceleration more than the pas-

sive-on case, although not as well as in the passive-off

case.

The controlled responses using the modulated homo-

geneous friction control algorithm indicate that this con-

trol algorithm is also quite effective at reducing both the

relative displacements and accelerations of the structure.

The maximum third floor displacement and interstory dis-

placement are reduces by an additional 20 and 23 , re-

spectively, over the best passive case. Furthermore, this

controller achieves a 10 reduction in the maximum abso-

lute acceleration as compared to the best passive case,

while using

a

significantly smaller peak force than all oth-

er control algorithms.

I

x

Figure

2

Diagram

of MR

Damper Implementation

5:

Conclusion

In

this paper

a

number of recently proposed, semi-ac-

tive control strategies were evaluated for use with the MR

damper through a numerical simulation. Here, a model of

a three-story structure was controlled using a single MR

damper. The results demonstrated that the performance of

the resulting controlled syistem and the requirements of the

control device are highly dependent on the control algo-

rithm employed. Each semi-active controller performed

noticeably better than the passive controllers in some way.

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5/5

Table

1

:El Centro Earthauake Peak Respons es

0.080

0.196

0.306

0.168

0.281

0.330

0.117

0.243

0.309

0.114

0.185

0.219

Control

Strategy

Uncontrolled

0.080

0.158

0.110

0.168

0.114

0.069

0.117

0.147

0.084

0.114

0.090

0.101

Passive-Off

Passive-On

Decentralized

Bang-Bang

Control

Lyapunov

Control

Clipped-

Optimal

Control

Modulated

Homogeneous

Friction

I

di

2 1

f

(cm) (cdsec ) (N)

(cm)

1400

0.211 0.211 420

0.357 0.153

0.455 0.103 258

1050

477

390

580

703

738 947

702

0.114 0.114 417

0.203 0.121 5 589

0.245 0.093

The largest reduction in the maximum acceleration was

achieved with the decentralized bang-bang approach. The

largest reduction in the third floor displacement and peak

interstory displacement was achieved with the clipped-op-

timal acceleration feedback algorithm. Furthermore, in

this example the modulated homogeneous friction ap-

proach performed quite well, achieving a significant

re-

duction in both the interstory displacement and the

maximum absolute acceleration.

Note that none of the control methods discussed here

requires a model for the M R damper, although a model is

important for system analysis. Algorithms that explicitly

incorporate actuator dynamics and control-structure inter-

action into the design process may offer additional perfor-

mance gains

[4].

Efforts are currently underway to

investigate this possibility.

Acknowledgment

This research is partially supported by National Sci-

ence Foundation Grant Nos. CMS 93-01584 CMS 95-

28083.

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Documents

1997 - A Comparison of Semi-Active Control Strategies for the MR Damper