ROBUST CONTROL OF HYSTERETIC BASE-ISOLATEDSTRUCTURES UNDER SEISMIC DISTURBANCES

Francesc Pozo, Jose RodellarCoDAlab, Universitat Politecnica de Catalunya, Comte d’Urgell, 187, 08036 Barcelona, Spain

francesc.pozo@upc.edu, jose.rodellar@upc.edu

Leonardo Acho, Ricardo GuerraCentro de Investigacion y Desarrollo de Tecnologıa Digital, Instituto Politecnico Nacional

Avenida del Parque, 1310, 22510 Tijuana, Baja California, Mexicoleonardo.acho@upc.edu, inge guerra@hotmail.com

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 eventof an earthquake. Taking advantage of discontinuous control theory, a static discontinuous active controlis developed using as a feedback only the measure of the velocity at the base. Moreover, due to that inmany engineering applications, accelerometers are the only devices that provide information available forfeedback, our velocity feedback controller could be easily extended by using just acceleration informationthrough 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. Robustnessperformance 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 upto 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 rigidbody with reduced inter-story drifts, as demandedby safety, and reduced absolute accelerations as re-lated to comfort requirements. Although the responsequantities of a fixed-base building are reduced sub-stantially through base isolation, the base displace-ment may be excessive, particularly during near-fieldground motions (Yang and Agrawal, 2002). Appli-cations of hybrid control systems consisting of activeor 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 abase-isolated structure. The first controller uses thevelocity 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). Dueto the fact that, in civil engineering applications, ac-celerometers are the most practically available sen-sors for feedback control, the second controller is anextension 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 theproblem statement using just velocity measurementsis 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 inSection 5. Finally, on Section 6 final comments arestated.

2 PROBLEM STATEMENT

Consider a basic forced vibration system governedby:

mx+ cx+Φ(x, t) = f (t)+u(t) , (1)

277

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 madewith inelastic rubber bearings; f (t) is the unknownexcitation force; and u(t) is the control force suppliedby an appropriate actuator.

In structure systems, f (t) = −mxg(t) is the exci-tation force, where xg (t) is the earthquake ground ac-celeration. The restoring force Φ can be representedby the Bouc-Wen model (Ikhouane et al., 2005) in thefollowing form:

Φ(x, t) = αKx(t)+(1−α)DKz(t) (2)

z = D−1 (Ax−β|x||z|n−1z−λx|z|n

)(3)

where Φ(x, t) can be considered as the superpositionof an elastic component αKx and a hysteretic com-ponent (1−α)DKz(t), in which the yield constantdisplacement is D > 0 and α ∈ [0,1] is the post- topre-yielding stiffness ratio. n≥ 1 is a scalar that gov-erns the smoothness of the transition from elastic toplastic 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 systemstructure 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) =−mxg is unknown but bounded; i.e., there exists aknown 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 boundon the internal dynamic variable z(t), i.e., |z(t)| ≤ρz, ∀t ≥ 0.

Assumption 1 is standard in control of hystereticsystems 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 whenthe 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 sectionssatisfy this control objective.

3 SEISMIC ATTENUATIONUSING ONLY VELOCITYFEEDBACK

Theorem 1 Consider the nonlinear system (1)-(3)subject to Assumptions 1-3. Then, the following con-trol law

u =−ρsgn(x0) (4)

solves the control objective if

ρ≥ (1−α)DKρz +F. (5)

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

m0x0 + c0x0 + k0x0 +Φ(x0, t) =−m0xg−ρsgn(x0)m0x0 + c0x0 +(k0 +αK)x0 =−ρsgn(x0)+∆(z, t)

(6)

where

∆(z, t) =−m0xg− (1−α)Dkz.

Then

|∆(z, t)| ≤ | f (t)|+ |(1−α)DKz|≤ F +(1−α)DK|z|≤ F +(1−α)DKρz = ρ1.

Given the Lyapunov function

V (x0, x0) =k0 +αK

2x2

0 +m0

2x2

0,

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its time derivative along the trajectories of the closed-loop system (1)-(3) and (4) yields

V (x0, x0) = (k0 +αK)x0x0 +m0x0x0

= x0 [(k0 +αK)x0 +m0x0]= x0 [−c0x0−ρsgn(x0)+∆(z, t)]

=−c0x20−ρx0sgn(x0)+ x0∆(z, t)

=−c0x20−ρ|x0|+ x0∆(z, t)

≤−c0x20−ρ|x0|+ |x0|ρ1

=−c0x20 +(ρ1−ρ)|x0|.

The choice of ρ ≥ ρ1 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 thisnon-smooth class of systems in the sense of Filippovhas been widely studied (Wu et al., 1998). It is worthnoting that non-smooth dynamic systems appearnaturally and frequently in many mechanical systems(Wu et al., 1998). Due to the fact that classicalsolution theories to ordinary differential equationsrequire vector fields to be at least Lipschitz con-tinuous, main difficulties with non-smooth systemsare that these systems fail the Lipschitz-continuousrequirement. However, if (a) the vector field ismeasurable and essentially bounded; (b) the solutionof the system is absolutely continuous; and (c) theLyapunov function V is continuous and positive def-inite and its time derivative V along the trajectoriesof the closed-loop system is continuous and negativesemi-definite, then the system under considerationhas a solution in the sense of Filippov and it is stablein the sense of Lyapunov (Wu et al., 1998). This isexactly our case.

Remark 2 The signum function in the control law inTheorem 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-likefunction such as

νδ(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 =−ρx0

|x0|+δ(7)

solves the control objective if

ρ≥ (1−α)DKρz +F

and δ is a sufficiently small positive scalar.

Proof. The time derivative of the Lyapunov func-tion

V (x0, x0) =k0 +αK

2x2

0 +m0

2x2

0,

along the trajectories of the closed-loop system (1)-(3) and (7) yields

V = (k0 +αK)x0x0 +m0x0x0

= x0 [(k0 +αK)x0 +m0x0]

= x0

[−c0x0−ρ

x0

|x0|+δ+∆(z, t)

]=−c0x2

0−ρx0x0

|x0|+δ+ x0∆(z, t)

=−c0x20−ρ

x20

|x0|+δ+ x0∆(z, t)

≤−c0x20 +ρ1|x0|−ρ

x20

|x0|+δ

=−c0x20− (ρ−ρ1)|x0|+ρ

(|x0|−

x20

|x0|+δ

).

The objective of guaranteeing global boundedness ofsolutions is equivalently expressed as rendering Vnegative outside a compact region. The choice ofρ≥ ρ1 and considering that

limδ→0

ρ

(|x0|−

x20

|x0|+δ

)= 0

guarantees the existence of a small compact regionD⊂ R2 (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 ATTENUATIONUSING ONLY ACCELERATIONFEEDBACK

Motivated by the fact that in many civil engineeringapplications accelerometers are the only devices thatprovide information available for feedback, Theorem2 (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 = x0 (9)

ROBUST CONTROL OF HYSTERETIC BASE-ISOLATED STRUCTURES UNDER SEISMIC DISTURBANCES

279

solves the control objective if

ρ≥ (1−α)DKρz +F.

Proof. This proof is straightforward by consider-ing direct integration of equation (9). �

Remark 3 In the practical implementation of thiscontrol 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+ x0, (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 inthe following Corollary.Corollary 2 Consider the nonlinear system (1)-(3)subject to Assumptions 1-3. Then, the following con-trol law

u =−ρυ

|υ|+δ(12)

υ =−συ+ x0 (13)solves the control objective if

ρ≥ (1−α)DKρz +F,

where σ > 0 and δ are sufficiently small positivescalar.

5 NUMERICAL SIMULATIONS

In order to investigate the efficiency of the pro-posed controllers, we set m = 156× 103 kg, K =6×106 N/m, c = 2×104 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 showsthe ground acceleration for this earthquake. The openloop base displacement can also be seen in Figure 2.It can be seen that ρ = 2 · 105 is an upper bound forthe expression (1−α)DKρz +F in equation (5).

Figures 3, 4 and 5 display the time histories of themotion 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 signalforce when the control law in equations (12)-(13) isused 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 worthnoting that, when σ = 0.1 in Figure 8, the results aresimilar to those in Figure 3.

0 5 10 15 20 25 30 35 40 45 50−3

−2

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0

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4

Time (s)

Gro

und

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lera

tion

(m/s

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El Centro earthquake

0 5 10 15 20 25 30 35 40 45 50−0.2

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−0.1

−0.05

0

0.05

0.1

0.15

0.2

Time (s)

Dis

plac

emen

t (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 structureshas been proposed. It has been shown that a sim-ple controller can fulfill the control objectives, usingjust velocity measurements or just acceleration infor-mation. Simulation results showed the good perfor-mance of the controllers. In civil engineering, thecontroller that just uses acceleration information is ofa great interest, due to the fact that accelerometers areeasily available. Also, the simplicity of the proposedcontrollers makes them attractive for a real implemen-tation.

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0 5 10 15 20 25 30 35 40 45 50−2

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0

1

2

3

4x 10

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Time (s)

Dis

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t (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

6x 10

5

time (s)

Forc

e (N

)

Control effort

Figure 3: Closed loop base displacement (top) and controlsignal force (bottom) with control law in equation (7) andparameters ρ = 2 ·106 and δ = 0.01.

0 5 10 15 20 25 30 35 40 45 50−2

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1

2

3

4x 10

−3

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Dis

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t (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

6x 10

5

time (s)

Forc

e (N

)

Control effort

Figure 4: Closed loop base displacement (top) and controlsignal force (bottom) with control law in equation (7) andparameters ρ = 2 ·106 and δ = 0.1.

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0 5 10 15 20 25 30 35 40 45 50−2

−1

0

1

2

3

4x 10

−3

Time (s)

Dis

plac

emen

t (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

2x 10

5

time (s)

Forc

e (N

)

Control effort

Figure 5: Closed loop base displacement (top) and controlsignal force (bottom) with control law in equation (7) andparameters ρ = 2 ·105 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

3.5x 10

−5

Time (s)

Dis

plac

emen

t (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

6x 10

4

time (s)

Forc

e (N

)

Control effort

Figure 6: Closed loop base displacement (top) and controlsignal force (bottom) with control law in equations (12)-(13) and parameters ρ = 2 ·106, δ = 0.01 and σ = 0.1.

0 5 10 15 20 25 30 35 40 45 50−2

−1

0

1

2

3

4x 10

−4

Time (s)

Dis

plac

emen

t (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

6x 10

4

time (s)

Forc

e (N

)

Control effort

Figure 7: Closed loop base displacement (top) and controlsignal force (bottom) with control law in equations (12)-(13) and parameters ρ = 2 ·106, δ = 0.1 and σ = 0.1.

0 5 10 15 20 25 30 35 40 45 50−4

−3

−2

−1

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2

3

4

5x 10

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t (m

)

Closed−loop base displacement

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−3

−2

−1

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2

3

4

5

6x 10

4

time (s)

Forc

e (N

)

Control effort

Figure 8: Closed loop base displacement (top) and controlsignal force (bottom) with control law in equations (12)-(13) and parameters ρ = 2 ·106, δ = 0.01 and σ = 1.

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