ROBUST CONTROL OF HYSTERETIC BASE-ISOLATED

STRUCTURES UNDER SEISMIC DISTURBANCES

Francesc Pozo, Jos

´

e Rodellar

CoDAlab, Universitat Polit

`

ecnica de Catalunya, Comte d’Urgell, 187, 08036 Barcelona, Spain

Leonardo Acho, Ricardo Guerra

Centro de Investigaci

´

on y Desarrollo de Tecnolog

´

ıa Digital, Instituto Polit

´

ecnico Nacional

Avenida del Parque, 1310, 22510 Tijuana, Baja California, Mexico

Keywords:

Base-isolated structures, seismic disturbances, active control.

Abstract:

The main objective of applying robust active control to base-isolated structures is to protect them in the event

of an earthquake. Taking advantage of discontinuous control theory, a static discontinuous active control

is developed using as a feedback only the measure of the velocity at the base. Moreover, due to that in

many engineering applications, accelerometers are the only devices that provide information available for

feedback, our velocity feedback controller could be easily extended by using just acceleration information

through a ﬁlter. The main contributions of this paper are: (a) a static velocity feedback controller design,

and (b) a dynamic acceleration feedback controller design, for seismic attenuation of structures. Robustness

performance is analyzed by means of numerical experiments using the 1940 El Centro earthquake.

1 INTRODUCTION

Base isolation has been widely considered as an ef-

fective technology to protect ﬂexible structures up

to eight storeys high against earthquakes. The con-

ceptual objective of the isolator is to produce a dy-

namic decoupling of the structure from its founda-

tion so that the structure ideally behaves like a rigid

body with reduced inter-story drifts, as demanded

by safety, and reduced absolute accelerations as re-

lated to comfort requirements. Although the response

quantities of a ﬁxed-base building are reduced sub-

stantially through base isolation, the base displace-

ment may be excessive, particularly during near-ﬁeld

ground motions (Yang and Agrawal, 2002). Appli-

cations of hybrid control systems consisting of active

or semi-active systems installed in parallel to base-

isolation bearings have the capability to reduce re-

sponse quantities of base-isolated structures more sig-

niﬁcantly than passive dampers (Ramallo et al., 2002;

Yang and Agrawal, 2002).

In this paper, two versions of a decentralized ro-

bust active control are developed and applied to a

base-isolated structure. The ﬁrst controller uses the

velocity at the base of the structure as feedback in-

formation, and it is analyzed via Lyapunov stability

techniques as proposed in (Luo et al., 2001). Due

to the fact that, in civil engineering applications, ac-

celerometers are the most practically available sen-

sors for feedback control, the second controller is an

extension of the ﬁrst one where just acceleration in-

formation is used. Performance of the proposed con-

trollers, for seismic attenuation, are evaluated by nu-

merical simulations using the 1940 El Centro earth-

quake (California, United States).

This paper is structured as follows. Section 2 de-

scribes the problem formulation. The solution to the

problem statement using just velocity measurements

is described in Section 3, meanwhile the solution em-

ploying only acceleration information is stated in Sec-

tion 4. Numerical simulations to analyze the perfor-

mance of both proposed controllers are presented in

Section 5. Finally, on Section 6 ﬁnal comments are

stated.

2 PROBLEM STATEMENT

Consider a basic forced vibration system governed

by:

m ¨x + c ˙x + Φ(x,t) = f (t) + u (t), (1)

277

Pozo F., Rodellar J., Acho L. and Guerra R. (2007).

ROBUST CONTROL OF HYSTERETIC BASE-ISOLATED STRUCTURES UNDER SEISMIC DISTURBANCES.

In Proceedings of the Fourth International Conference on Informatics in Control, Automation and Robotics, pages 277-282

DOI: 10.5220/0001639202770282

Copyright

c

SciTePress

Base

Active Controller

Isolator

Foundation

Earthquake

m

c

f(t) = -ma(t)

u(t)

Φ

Figure 1: Building structure with hybrid control system (up)

and physical model (down).

where m is the mass; c is the damping coefﬁcient; Φ

is the restoring force characterizing the hysteretic be-

havior of the isolator material, which is usually made

with inelastic rubber bearings; f (t) is the unknown

excitation force; and u(t) is the control force supplied

by an appropriate actuator.

In structure systems, f (t) = −m ¨x

g

(t) is the exci-

tation force, where ¨x

g

(t) is the earthquake ground ac-

celeration. The restoring force Φ can be represented

by the Bouc-Wen model (Ikhouane et al., 2005) in the

following form:

Φ(x,t) = αKx (t) + (1 − α) DKz (t) (2)

˙z = D

−1

A ˙x − β| ˙x||˙z|

n−1

z − λ ˙x|z|

n

(3)

where Φ(x,t) can be considered as the superposition

of an elastic component αKx and a hysteretic com-

ponent (1 − α)DKz(t), in which the yield constant

displacement is D > 0 and α ∈ [0, 1] is the post- to

pre-yielding stiffness ratio. n ≥ 1 is a scalar that gov-

erns the smoothness of the transition from elastic to

plastic response and K > 0. The hysteretic part in (2)

involves an internal dynamic (3) which is unmeasur-

able, and thus inaccessible for seismic control design.

A schematic description of the base-isolated system

structure and its physical model are displayed in Fig.

1.

The following assumptions are stated for system (1)-

(3):

Assumption 1 The acceleration disturbance f (t) =

−m ¨x

g

is unknown but bounded; i.e., there exists a

known constant F such that | f (t)| ≤ F, ∀t ≥ 0.

Assumption 2 In the event of an earthquake, it is as-

sumed that z(0) = 0 in equation (1) and that the struc-

ture is at rest; i.e., x(0) = ˙x(0) = 0.

Assumption 3 There exists a known upper bound

on the internal dynamic variable z(t), i.e., |z(t)| ≤

¯

ρ

z

, ∀t ≥ 0.

Assumption 1 is standard in control of hysteretic

systems or base-isolated structures (Ikhouane et al.,

2005). Assumption 2 has a physical meaning be-

cause it is assumed that the structure is at rest when

the earthquake strikes it. The upper bound in z(t)

expressed in Assumption 3 is computable, indepen-

dently on the boundedness of x(t) by invoking Theo-

rem 1 in (Ikhouane et al., 2005).

Control objective: Our objective is to design a ro-

bust controller for system (1) such that, under earth-

quake attack, the trajectories of the closed-loop re-

main bounded.

To this end, the theorems in the following sections

satisfy this control objective.

3 SEISMIC ATTENUATION

USING ONLY VELOCITY

FEEDBACK

Theorem 1 Consider the nonlinear system (1)-(3)

subject to Assumptions 1-3. Then, the following con-

trol law

u = −ρsgn( ˙x

0

) (4)

solves the control objective if

ρ ≥ (1 − α)DK

¯

ρ

z

+ F. (5)

Proof. The closed-loop system (1)-(3) and (4)

yields

m

0

¨x

0

+ c

0

˙x

0

+ k

0

x

0

+ Φ(x

0

,t) = −m

0

¨x

g

− ρsgn( ˙x

0

)

m

0

¨x

0

+ c

0

˙x

0

+ (k

0

+ αK)x

0

= −ρsgn( ˙x

0

) + ∆(z, t)

(6)

where

∆(z,t) = −m

0

¨x

g

− (1 − α)Dkz.

Then

|∆(z,t)| ≤ | f (t)| + |(1 − α)DKz|

≤ F + (1 − α)DK|z|

≤ F + (1 − α)DK

¯

ρ

z

= ρ

1

.

Given the Lyapunov function

V (x

0

, ˙x

0

) =

k

0

+ αK

2

x

2

0

+

m

0

2

˙x

2

0

,

ICINCO 2007 - International Conference on Informatics in Control, Automation and Robotics

278

its time derivative along the trajectories of the closed-

loop system (1)-(3) and (4) yields

˙

V (x

0

, ˙x

0

) = (k

0

+ αK)x

0

˙x

0

+ m

0

˙x

0

¨x

0

= ˙x

0

[(k

0

+ αK)x

0

+ m

0

¨x

0

]

= ˙x

0

[−c

0

˙x

0

− ρsgn( ˙x

0

) + ∆(z, t)]

= −c

0

˙x

2

0

− ρ ˙x

0

sgn( ˙x

0

) + ˙x

0

∆(z,t)

= −c

0

˙x

2

0

− ρ| ˙x

0

| + ˙x

0

∆(z,t)

≤ −c

0

˙x

2

0

− ρ| ˙x

0

| + | ˙x

0

|ρ

1

= −c

0

˙x

2

0

+ (ρ

1

− ρ)| ˙x

0

|.

The choice of ρ ≥ ρ

1

makes

˙

V negative semideﬁnite,

as we wanted to show.

Remark 1 (on solution of non-smooth systems)

The closed-loop system (6) has a non-smooth right-

hand side, the signum function. Solutions to this

non-smooth class of systems in the sense of Filippov

has been widely studied (Wu et al., 1998). It is worth

noting that non-smooth dynamic systems appear

naturally and frequently in many mechanical systems

(Wu et al., 1998). Due to the fact that classical

solution theories to ordinary differential equations

require vector ﬁelds to be at least Lipschitz con-

tinuous, main difﬁculties with non-smooth systems

are that these systems fail the Lipschitz-continuous

requirement. However, if (a) the vector ﬁeld is

measurable and essentially bounded; (b) the solution

of the system is absolutely continuous; and (c) the

Lyapunov function V is continuous and positive def-

inite and its time derivative

˙

V along the trajectories

of the closed-loop system is continuous and negative

semi-deﬁnite, then the system under consideration

has a solution in the sense of Filippov and it is stable

in the sense of Lyapunov (Wu et al., 1998). This is

exactly our case.

Remark 2 The signum function in the control law in

Theorem 1 –common in sliding mode control theory–

produces chattering (Utkin, 1982; Edwards and Spur-

geon, 1998). One way to avoid chattering is to re-

place the signum function by a smooth sigmoid-like

function such as

ν

δ

(s) =

s

|s| + δ

,

where δ is a sufﬁciently small positive scalar (Ed-

wards and Spurgeon, 1998).

Consequently, the following Corollary is stated:

Corollary 1 Consider the nonlinear system (1)-(3)

subject to Assumptions 1-3. Then, the following con-

trol law

u = −ρ

˙x

0

| ˙x

0

| + δ

(7)

solves the control objective if

ρ ≥ (1 − α)DK

¯

ρ

z

+ F

and δ is a sufﬁciently small positive scalar.

Proof. The time derivative of the Lyapunov func-

tion

V (x

0

, ˙x

0

) =

k

0

+ αK

2

x

2

0

+

m

0

2

˙x

2

0

,

along the trajectories of the closed-loop system (1)-

(3) and (7) yields

˙

V = (k

0

+ αK)x

0

˙x

0

+ m

0

˙x

0

¨x

0

= ˙x

0

[(k

0

+ αK)x

0

+ m

0

¨x

0

]

= ˙x

0

−c

0

˙x

0

− ρ

˙x

0

| ˙x

0

| + δ

+ ∆(z,t)

= −c

0

˙x

2

0

− ρ ˙x

0

˙x

0

| ˙x

0

| + δ

+ ˙x

0

∆(z,t)

= −c

0

˙x

2

0

− ρ

˙x

2

0

| ˙x

0

| + δ

+ ˙x

0

∆(z,t)

≤ −c

0

˙x

2

0

+ ρ

1

| ˙x

0

| − ρ

˙x

2

0

| ˙x

0

| + δ

= −c

0

˙x

2

0

− (ρ − ρ

1

)| ˙x

0

| + ρ

| ˙x

0

| −

˙x

2

0

| ˙x

0

| + δ

.

The objective of guaranteeing global boundedness of

solutions is equivalently expressed as rendering

˙

V

negative outside a compact region. The choice of

ρ ≥ ρ

1

and considering that

lim

δ→0

ρ

| ˙x

0

| −

˙x

2

0

| ˙x

0

| + δ

= 0

guarantees the existence of a small compact region

D ⊂ R

2

(depending on δ) such that

˙

V is negative out-

side this set. This implies that all the closed-loop tra-

jectories remain bounded, as we wanted to show.

4 SEISMIC ATTENUATION

USING ONLY ACCELERATION

FEEDBACK

Motivated by the fact that in many civil engineering

applications accelerometers are the only devices that

provide information available for feedback, Theorem

2 (below) presents a control law based on equation (4)

where only acceleration information is required.

Theorem 2 Consider the nonlinear system (1)-(3)

subject to Assumptions 1-3. Then, the following con-

trol law

u = −ρsgn(υ) (8)

˙

υ = ¨x

0

(9)

ROBUST CONTROL OF HYSTERETIC BASE-ISOLATED STRUCTURES UNDER SEISMIC DISTURBANCES

279

solves the control objective if

ρ ≥ (1 − α)DK

¯

ρ

z

+ F.

Proof. This proof is straightforward by consider-

ing direct integration of equation (9).

Remark 3 In the practical implementation of this

control law, ν may drift due to unmodeled dynam-

ics, measure errors and disturbance. To avoid this,

the following σ-modiﬁcation (Ioannou and Kokotovic,

1983; Koo and Kim, 1994) can be used,

u = −ρsgn(υ), (10)

˙

υ = −σν + ¨x

0

, (11)

where σ is a positive constant.

As in the previous Section, a smooth version of

the control law in equations (10)-(11) is considered in

the following Corollary.

Corollary 2 Consider the nonlinear system (1)-(3)

subject to Assumptions 1-3. Then, the following con-

trol law

u = −ρ

υ

|υ| + δ

(12)

˙

υ = −συ + ¨x

0

(13)

solves the control objective if

ρ ≥ (1 − α)DK

¯

ρ

z

+ F,

where σ > 0 and δ are sufﬁciently small positive

scalar.

5 NUMERICAL SIMULATIONS

In order to investigate the efﬁciency of the pro-

posed controllers, we set m = 156 × 10

3

kg, K =

6 × 10

6

N/m, c = 2 × 10

4

Ns/m, α = 0.6, D = 0.6 m,

λ = 0.5, β = 0.1, n = 3, and A = 1 (Ikhouane et al.,

2005). A set of numerical experiments was per-

formed on the system using information recorded dur-

ing the 1940 El Centro earthquake. Figure 2 shows

the ground acceleration for this earthquake. The open

loop base displacement can also be seen in Figure 2.

It can be seen that ρ = 2 · 10

5

is an upper bound for

the expression (1 − α)DK

¯

ρ

z

+ F in equation (5).

Figures 3, 4 and 5 display the time histories of the

motion of the base and the control signal force for dif-

ferent values of ρ and δ, when the control law in equa-

tion (7) is used. In an equivalent manner, the time his-

tories of the motion of the base and the control signal

force when the control law in equations (12)-(13) is

used are depicted in Figures 6, 7 and 8. In both cases,

the controlled base displacements are signiﬁcantly re-

duced compared to the uncontrolled case. It is worth

noting that, when σ = 0.1 in Figure 8, the results are

similar to those in Figure 3.

0 5 10 15 20 25 30 35 40 45 50

−3

−2

−1

0

1

2

3

4

Time (s)

Ground acceleration (m/s

2

)

El Centro earthquake

0 5 10 15 20 25 30 35 40 45 50

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Time (s)

Displacement (m)

Open−loop base displacement

Figure 2: 1940 El Centro earthquake, ground acceleration

(top); open loop base displacement (bottom).

6 CONCLUSION

A robust control scheme to attenuate the conse-

quences of seismic events on base-isolated structures

has been proposed. It has been shown that a sim-

ple controller can fulﬁll the control objectives, using

just velocity measurements or just acceleration infor-

mation. Simulation results showed the good perfor-

mance of the controllers. In civil engineering, the

controller that just uses acceleration information is of

a great interest, due to the fact that accelerometers are

easily available. Also, the simplicity of the proposed

controllers makes them attractive for a real implemen-

tation.

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0 5 10 15 20 25 30 35 40 45 50

−2

−1

0

1

2

3

4

x 10

−4

Time (s)

Displacement (m)

Closed−loop base displacement

0 5 10 15 20 25 30 35 40 45 50

−4

−3

−2

−1

0

1

2

3

4

5

6

x 10

5

time (s)

Force (N)

Control effort

Figure 3: Closed loop base displacement (top) and control

signal force (bottom) with control law in equation (7) and

parameters ρ = 2 · 10

6

and δ = 0.01.

0 5 10 15 20 25 30 35 40 45 50

−2

−1

0

1

2

3

4

x 10

−3

Time (s)

Displacement (m)

Closed−loop base displacement

0 5 10 15 20 25 30 35 40 45 50

−4

−3

−2

−1

0

1

2

3

4

5

6

x 10

5

time (s)

Force (N)

Control effort

Figure 4: Closed loop base displacement (top) and control

signal force (bottom) with control law in equation (7) and

parameters ρ = 2 · 10

6

and δ = 0.1.

ROBUST CONTROL OF HYSTERETIC BASE-ISOLATED STRUCTURES UNDER SEISMIC DISTURBANCES

281

0 5 10 15 20 25 30 35 40 45 50

−2

−1

0

1

2

3

4

x 10

−3

Time (s)

Displacement (m)

Closed−loop base displacement

0 5 10 15 20 25 30 35 40 45 50

−1.5

−1

−0.5

0

0.5

1

1.5

2

x 10

5

time (s)

Force (N)

Control effort

Figure 5: Closed loop base displacement (top) and control

signal force (bottom) with control law in equation (7) and

parameters ρ = 2 · 10

5

and δ = 0.01.

0 5 10 15 20 25 30 35 40 45 50

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

x 10

−5

Time (s)

Displacement (m)

Closed−loop base displacement

0 5 10 15 20 25 30 35 40 45 50

−4

−3

−2

−1

0

1

2

3

4

5

6

x 10

4

time (s)

Force (N)

Control effort

Figure 6: Closed loop base displacement (top) and control

signal force (bottom) with control law in equations (12)-

(13) and parameters ρ = 2 · 10

6

, δ = 0.01 and σ = 0.1.

0 5 10 15 20 25 30 35 40 45 50

−2

−1

0

1

2

3

4

x 10

−4

Time (s)

Displacement (m)

Closed−loop base displacement

0 5 10 15 20 25 30 35 40 45 50

−4

−3

−2

−1

0

1

2

3

4

5

6

x 10

4

time (s)

Force (N)

Control effort

Figure 7: Closed loop base displacement (top) and control

signal force (bottom) with control law in equations (12)-

(13) and parameters ρ = 2 · 10

6

, δ = 0.1 and σ = 0.1.

0 5 10 15 20 25 30 35 40 45 50

−4

−3

−2

−1

0

1

2

3

4

5

x 10

−5

Time (s)

Displacement (m)

Closed−loop base displacement

0 5 10 15 20 25 30 35 40 45 50

−4

−3

−2

−1

0

1

2

3

4

5

6

x 10

4

time (s)

Force (N)

Control effort

Figure 8: Closed loop base displacement (top) and control

signal force (bottom) with control law in equations (12)-

(13) and parameters ρ = 2 · 10

6

, δ = 0.01 and σ = 1.

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282