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Dynamic Force Control with Hydraulic Actuators Using Added

Compliance and Displacement Compensation

By

Mettupalayam V. Sivaselvan University of Colorado, Boulder

Andrei M. Reinhorn, Xiaoyun Shao,

and Scot Weinreber University at Buffalo

Center for Fast Hybrid Testing Department of Civil Environmental and Architectural Engineering University of Colorado UCB 428 Boulder, Colorado 80309-0428

October 2008

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS

Earthquake Engng Struct. Dyn. 2007; 00:1–10 Prepared using eqeauth.cls [Version: 2002/11/11 v1.00]

Dynamic force control with hydraulic actuators

using added compliance and displacement compensation

Mettupalayam V. Sivaselvan1∗, Andrei M. Reinhorn2, Xiaoyun Shao2

and Scot Weinreber2

1 Department of Civil, Environmental and Architectural Engineering, University of Colorado at Boulder,

Boulder, CO 80309

2 Department of Civil, Structural and Environmental Engineering, University at Buffalo, Buffalo, NY 14260

SUMMARY

A new approach to dynamic force control of mechanical systems, applicable in particular to frame

structures, over frequency ranges spanning their resonant frequencies is presented. This approach

is implemented using added compliance and displacement compensation. Hydraulic actuators are

inherently velocity sources, that is, an electrical signal regulates their velocity response. Such systems

are therefore by nature high-impedance (mechanically stiff) systems. In contrast for force control, a

force source is required. Such a system logically would have to be a low-impedance (mechanically

compliant) system. This is achieved by intentionally introducing a flexible mechanism between the

∗Correspondence to: 428 UCB, University of Colorado at Boulder, Boulder, CO 80309

Email: [email protected], Phone: (303)735-0925, FAX: (303)492-7317

Contract/grant sponsor: George E. Brown Network for Earthquake Engineering Simulation, National Science

Foundation; contract/grant number: #CMS-0086611 and #CMS-0086612

Received

Copyright c© 2007 John Wiley & Sons, Ltd. Revised

2 M. V. SIVASELVAN ET. AL.

actuator, and the structure to be excited. In addition, in order to obtain force control over frequencies

spanning the structure’s resonant frequency, a displacement compensation feedback loop is needed.

The actuator itself operates in closed-loop displacement control. The theoretical motivation as well

as the laboratory implementation of the above approach is discussed along with experimental results.

Having achieved a means of dynamic force control, it can be applied to various experimental seismic

simulation techniques such as the Effective Force Method and the Real-time Dynamic Hybrid Testing

Method. Copyright c© 2007 John Wiley & Sons, Ltd.

key words: Dynamics Force Control, Hydraulic Actuators, Natural Velocity Feedback, Smith

Predictor, Advanced Seismic Testing

1. INTRODUCTION

Advanced seismic testing techniques such as the effective force method [3] and forms of real-

time dynamic hybrid testing [15] require the implementation of dynamic force control in

hydraulic actuators. Dynamic force control with hydraulic actuators is however a challenging

problem. By its physical nature, a hydraulic actuator is a velocity source, i.e., a given controlled

flow rate into the actuator results in a certain velocity. Moreover, hydraulic actuators are

typically designed for good position control, i.e., to move heavy loads quickly and accurately.

They are therefore by construction, high impedance (mechanically stiff) systems [12]. In

contrast a force source is required for force control. Such a system logically would have to

be a low-impedance (mechanically compliant) system.

Force control with hydraulic actuators is associated with many problems. Actuators designed

for position control have stiff oil columns, making force control very sensitive to control

Copyright c© 2007 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2007; 00:1–10

Prepared using eqeauth.cls

DYNAMIC FORCE CONTROL 3

parameters often leading to instability. Moreover friction, stick-slip, breakaway forces on seals,

backlash etc. cause noise in the force measurement, making force a difficult quantity to control.

Several strategies have been introduced to work around this problem. For instance, a dual

compensation scheme [11] uses a primary displacement feedback loop with force as a secondary

tracking feedback. This scheme also supports features such as acceleration compensation to

overcome some of the effects that distort the force measurements. In robotics, the impedance

control strategy has been employed wherein the force-displacement relationship is controlled

at the actuator interface [7, 19]. Pratt et. al. [14] have used the idea of “series elastic actuators”

where a flexible mechanism is intentionally introduced between the actuator and the point of

application of force, along with force feedback. They applied this to non-resonant systems.

Furthermore, in force control the dynamics of the structure on which force is applied, is

coupled in a feedback system with the dynamics of the actuator, resulting in a natural velocity

feedback. When the structure is resonant, this results in a set of complex conjugate zeros of the

open loop transfer function. Shield et. al. [3, 17] in their work on the effective force method,

compensate for this effect by using velocity feedforward. It was also recognized by Conrad

and Jensen [1], that closed-loop control with force feedback is ineffective without velocity

feedforward, or full state feedback.

In this paper, a new approach to dynamic force control is presented, in which a compliance in

the form of a spring is intentionally introduced between the actuator and the structure, and a

displacement feedforward compensation is used. The method does not use direct force feedback.

It also allows for an added physical design parameter in the control system, namely the stiffness

of the added compliance. In the following, a standard linearized dynamic model of a hydraulic

actuator is first presented. The natural velocity feedback problem and the solution of Shield et.




al. [3] are then discussed side by side to emphasize the differences and commonalities with the

approach presented in this paper. The motivation behind the proposed solution using added

compliance and displacement compensation is then discussed. The analysis of the proposed

solution and some experimental results are then presented.

2. LINEAR MODEL OF A SERVO-HYDRAULIC ACTUATOR

A hydraulic actuator driving a single degree of freedom structure is shown in Figure 1. The

analysis is this paper is based on linear models of the actuator and of the structure. For this,

the dynamics of the actuator are linearized about the equilibrium point at the mid-stroke of

the actuator. The linearized equations are standard (see for example [9, 2, 5, 18]) and are given

by

xp = vp

vp =Ap

M∆P − ω2

stxp − 2ζstωstvp

∆P = 2κ

ApL(−Apvp − γ1∆P + γ2xv)

xv = −

1

τv

+Kv

τv

u

(1)

Here, xp and vp are respectively the displacement of the SDOF structure, ∆P is the differential

pressure between the actuator chambers, xv is the valve spool displacement, M is the combined

mass of the actuator piston and the SDOF system, L is half the stroke of the actuator, Ap is

the area of the actuator piston, τv is the servovalve time constant, Kv is the servovalve gain,

κ is the bulk modulus of the oil, γ1 is a dissipative constant that depends on the chamber and

valve leakage flows, γ2 is a gain coefficient, ωst and ζst are the natural frequency and damping

ratio of the SDOF structure and u is the control input to the servovalve. A block diagram




Supply

Return

PR

~ 0

PS

1

2 3

4

MP1 P2

xv

xp kst

cst

Figure 1. Model of a hydraulic actuator driving a SDOF structure

Σ12 1

pA

C s γ+

2 2

1

2 st st st

s

M s sζ ω ω+ +

Ap

+

-

NaturalVelocity

Feedback

ValveCommand, u

Servovalve

Applied Force, f

Actuator

Structure

Flow

1v

K

sτ +

Figure 2. Block diagram representation of the linear model of equation (1).h

C12 =ApL

2κ, K = Kvγ2

i

model of this linear system is shown in Figure 2. The quantity

ωoil =

√

2κAp

LM(2)

is referred to as the oil column frequency. This is the imaginary part of a complex conjugate

eigenvalue pair of the linearization, in the absence of a structure stiffness.




3. NATURAL VELOCITY FEEDBACK AND THE ASSOCIATED CONTROL PROBLEM

It can be seen from the third part of equations (1), that the velocity of the mass affects the

rate of change of the differential pressure. This feedback can also be seen in the block diagram

of Figure 2. This has been termed natural velocity feedback [3]. If the dissipation related to

leakage flows, γ1 is assumed to be zero, then the resulting transfer function Huf from the

control input u to the applied force f is given by

Huf =K

s (τvs + 1)

s2 + 2ζstωsts + ω2st

s2 + 2ζstωsts + ω2st + ω2

oil

(3)

It can be seen that this transfer function has a complex conjugate pair of zeros corresponding

to the natural frequency and damping ratio of the SDOF structure. This implies that the force

applied on the structure at this frequency becomes small. Feedback control using for example a

PID controller does not improve the performance because these zeros persist in the closed-loop

transfer function also. Therefore, additional control strategies are necessary.

3.1. Strategy of Velocity Feedback Compensation (Shield et. al. [3, 17])

It can be seen that there is a negative feedback of velocity at the summing junction in Figure

1. If we can add a positive feedback at this junction of the same amount, then the effect

of the natural velocity feedback can be nullified. But since this is a physical junction that

in inaccessible, the strategy of Shield et. al. [3, 17] is to add this positive feedback to the

valve command. However, this signal has to now be preconditioned by the pseudo-inverse of

the servovalve transfer function. This is done using a lead-lag compensator. In addition force

feedback is also used. Figure 1 shows the resulting control strategy [3].




Σ K Σ12

pA

C s

Ap

+

+

+-

-

Desired Force

NaturalVelocity

Feedback

ValveCommand

Servovalve

VelocityCompensation

Achieved Force

Actuator

Structure

ForceFeedback

Flow

2 2

1

2 st st st

s

M s sζ ω ω+ +Compensator

Figure 3. Block diagram showing velocity feedforward correction loop

4. MOTIVATION FOR SOLUTION BASED ON ADDED COMPLIANCE AND

DISPLACEMENT COMPENSATION

It is known from experience that hydraulic actuators are more conveniently tuned in closed-loop

position control, than in force control. It is therefore suggested to indirectly control force by

controlling position. To do this, a compliance, a spring of stiffness kLC , is introduced between

the actuator and the structure. In this section, for simplicity of illustration, the effect of the

reaction force from the spring on the actuator is ignored, i.e., perfect disturbance rejection

is assumed. Perfect tracking is also assumed over all frequencies of interest. The full linear

dynamics of the actuator is however considered in the analysis in section 5. First, the scenario

the scenario of applying a force f on a rigid structure is considered as shown in Figure 4. It

is easily seen that to apply a force f , the actuator piston needs to move an amount f/kLC .

Thus the actuator can be operated in closed-loop position control, and be commanded to

the position f/kLC . If the structure were not rigid but flexible, then the applied force would

cause it to displace by an amount xst. Thus the actuator needs to be commanded to the

position f/kLC + xst. This leads to the need for displacement compensation. The structure

displacement xst may be obtained by from a model of the structure, or by measurement. These




Position Command

Measured Force, f

Rigid Structure

Actuator in Closed-loop

Position Control

Desired Force, f

1 / kLC

Added Compliance, kLC

Figure 4. Applying a desired force on a rigid structure by controlling the position of the actuator

possibilities are shown in Figure 5. It will be seen later that a mix of the two approaches leads

to the Smith Predictor approach. In addition, since the assumptions of perfect tracking and

perfect disturbance rejection in the above discussion are not realistic, additional compensation

is needed for the dynamics of the actuator. This is presented in section 5 below. However, first

the relationship of this approach to that of Shield et. al. is shown.

4.1. Comparison of the Proposed Approach to Velocity Compensation

The relationship of the proposed approach to the velocity feedback compensation strategy of

Shield et. al. [3, 17] can be shown by rearranging terms in the block diagram in Figure 3.

Factoring Aps suitably in Figure 3, the block diagram in Figure 6 is obtained. Comparing the

block diagrams in Figures 6 and 5(c), it is seen that the in the absence of added compliance,

the oil column behaves as a spring providing the compliance required for force control. Relative

deformation occurs across this spring and force is applied through it. However, the compliance

of the oil column spring is fixed for a given actuator. In the approach proposed here, this

compliance becomes an additional physical design parameter for the control system.




Flexible Structure

Position Command


Measured Force, f


Position Control

1 / kLC

Structure Model

+

+

+

+

Desired Force, f

(a) Using a model to obtain the structure displacement

Flexible Structure

Position Command


Measured Force, f


Position Control

1 / kLC+

++

+

Desired Force, f

Structure Displacement, xst

(b) Using measured structure displacement

Σ Σ

( )2 2

1

2 st st stM s sζ ω ω+ +

+

+

+

-

Desired Force

Displacement Command

Displacement Compensation

Achieved Force

Added Compilance

Structure

Actuator in Position Control

ActuatorDisplacement

StructureDisplacement

Compensator

kLC1/kLC

(c) Block diagram representation of (b)

Figure 5. Applying a desired force on a flexible structure

5. ANALYSIS OF THE PROPOSED CONTROL SOLUTION

The analysis of the proposed strategy for dynamic force control with added compliance and

displacement compensation is based on a linearized model, which is first presented.




Σ Σ

( )2 2

1


+

+

+-

-

Desired Force

ValveCommand

Servovalve + Actuator

Displacement Compensation

Achieved Force

Oil spring

Structure

ForceFeedback

p

K

A sActuator

Displacement

StructureDisplacement

Compensator

2 pA

L

κ

Figure 6. Refactoring of block diagram in Figure 3

5.1. Linear Modeling

Modifying the model in equation (1) suitably, the linearized model of the actuator and the

structure with the added compliance is obtained as

xp = vp

vp =Ap

mp

∆P − kLC (xp − xst)

∆P = 2κ

ApL(−Apvp − γ1∆P + γ2xv)

xv = −

1

τv

+Kv

τv

u

xst = vst

vst = −ω2stxp − 2ζstωstvp − kLC (xst − xp)

(4)

Here, mp is the mass of the piston, xxt and vst are the displacement and velocity of the

structure (which are now different from those of the actuator piston because of the added

compliance), kLC is the stiffness of the added compliance and the other symbols are as defined

before. The block diagram representation of this system along with the position controller C1

and the displacement feedforward compensator C2 are shown in Figure 7. In the figure, A1

and A2 are actuator transfer functions respectively from the valve command to the actuator




Desired Force, f 1/kLC C2Σ

+

+ Σ-

+C1 A1 Σ+

-Σ kLC

S

+

-

A2

Position Command

xp

xst

Applied Force

Figure 7. Block diagram of the linear model of the actuator and structure with added compliance and

the inner and outer loop controllers

Desired Force, f 1/kLC C2Σ

+

+ Σ-

+C1 Σ kLC

S

+

-

Position Command

xp

xst

Applied Force

( )( )

1

2

1

1LC

LC

A k S

k A S

++ +

Figure 8. Block diagram of Figure 7 rearranged

displacement, and from the force on the piston to the actuator displacement. These are given

by

A1 =4Kκ

mpLs (τvs + 1) (s2 + 2ζaωoils + ω2oil)

A2 =s + 2ζaωoil

mps (s2 + 2ζaωoils + ω2oil)

(5)

where ωoil =√

2Apκ

mpLis the oil column frequency, ζa = γ1

√

mpκ

a3pL

is the actuator damping ratio,

and S is the transfer function of the SDOF structure,

S =1

mst (s2 + 2ζstωsts + ω2st)

(6)

The block diagram in Figure 7 can be rearranged as shown in Figure 8. The block diagram

consists of an inner loop with controller C1 whose role is to track the position command, and

an outer loop which provides displacement feedforward compensation. The role of the C2 is to

compensate for the dynamics of the inner loop. The inner loop dynamics and the controller C1




Σ-

+C1

Position

Command xp( )

( )1

2

1

1LC

LC

A k S

k A S

++ +

Figure 9. Inner Loop

are presented in section 5.2. The outer loop and the compensator C2 are discussed in section

6.

5.2. Inner Loop

The inner loop is shown in Figure 9. It can be seen that a more compliant spring, i.e. a lower kLC

relative to the structure stiffness and the oil column stiffness, results in reducing the influence

of the structure dynamics S, and of the effect of the reaction force A2 on the dynamics of

the actuator A1. Physically, this can be interpreted as the compliant spring “isolating” the

dynamics of the actuator from that of the structure for displacement tracking. The role of the

controller C1 is to track the position command. For this purpose, a proprietary control system,

typically implementing a PID control can be used. The control system can be tuned with the

structure connected to the actuator through the spring. Experience shows that the controller

C1 can be tuned in most cases so that the inner loop dynamics has a nearly flat frequency

response magnitude with a linearly increasing phase lag over the bandwidth of interest. The

inner loop dynamics can therefore be modeled reasonably as a pure time delay. This approach

is used in modeling the inner loop dynamics.




5.3. Response without Outer-Loop Compensation

If an explicit outer loop compensator is not used, i.e. C2 is set equal to 1, then the transfer

function from the desired force to the measured force is given by

fachieved

fdesired

=s2 + 2ζstωsts + ω2

st

s2 + 2ζstωsts + ω2st + kLC

mst(1 − IL)

(7a)

where IL is the inner loop transfer function. If the inner loop dynamics is modeled by a pure

time-delay, and a first order Taylor series approximation of the delay is used (i.e., (1−IL) ≈ τs),

then this transfer function reduces to

fachieved

fdesired

=s2 + 2ζstωsts + ω2

st

s2 +(

2ζstωst + kLC

mstτd

)

s + ω2st

(7b)

where τd is the time-delay of the inner loop dynamics. It is seen that the delay, to a first order

approximation, has effect of increasing the damping of the poles of the transfer function. For

a lightly damped structure, lightly damped zeros still exist in the transfer function. These

zeros manifest as a drop in the frequency response magnitude at the resonant frequency of

the SDOF structure as shown in Figure 15(a). This necessitates the design of the outer loop

compensator C2.

6. OUTER LOOP COMPENSATOR DESIGN

Motivated by the fact that the inner loop dynamics can be reasonably modeled as a pure time-

delay, we consider the Smith predictor is considered as an approach to design the compensator

C2 of Figure 8. The Smith predictor was developed as a time-delay compensation algorithm in

chemical process control [6]. It is however applicable to compensate for other types of dynamics

as well. In the following, the basic idea of the Smith predictor is first reviewed, followed by a

description of how it can be used to compensate for the inner loop dynamics.




6.1. The Smith Predictor

The basic idea of the Smith predictor is described by constructing it based on motivation.

Figure 10(a) shows a standard feedback control system where a controller C has been designed

for the plant P in such a way that the closed loop system has certain desired characteristics. The

closed-loop transfer function is PC1+PC

. However, the control input cannot be applied directly

to the plant, but has to be applied through an actuator A. The dynamics of the actuator

may be thought of as “undesirable” dynamics in the feedback path. In order to regain the

original closed loop structure, feedback is obtained from a model of the plant, P instead of

from the plant itself as shown in Figure 10(b). However due to modeling error, the feedback

obtained from the model of the plant P will not be the same as what would have been obtained

from the actual plant P in the absence of the undesirable dynamics. Therefore, an additional

error feedback is used as shown in Figure 10(c). Here, A is the transfer function model of the

actuator dynamics. This leads to the Smith Predictor architecture. It can be seen that if the

models were exact, i.e. A = A and P = P , then the transfer function from reference to output

is PC1−PC

A, and the Smith Predictor has the effect of moving the “undesirable” dynamics out

of the feedback loop. The Smith Predictor is also intimately related to the Internal Model

Control idea (see for example, [10]).

6.2. Smith Predictor for Compensation of Inner Loop Dynamics

As discussed in section 4, a desired force f is applied on the SDOF structure by imposing a

displacement of f/kLC +xst to the end of the added compliance. Thus the feedback structure is

as shown in Figure 11, corresponding to the idea depicted in Figure 5(b). This is the “desired”

feedback structure corresponding to Figure 10(a). However in reality, also present in this




PCΣ-

+ outputreference

(a) Standard feedback control system

outputΣ

-

+C

reference

A P

P

(b) Using feedback from the model to avoid undesirable

dynamics A in the feedback path

outputΣ-

+C

referenceA P

P A

Σ-

+

Σ-+

error

(c) The Smith Predictor

Figure 10. The concept of the Smith Predictor

Σ Σ

( )2 2

1


+

+

+

-

Desired Force, f

Achieved Force

Added Compilance

Structure

kLC1/kLC

xst

stLC

fx

k+

1

Figure 11. Desired feedback structure for displacement compensation




feedback loop is the “undesirable” dynamics of the inner loop as shown in Figure 8. The

corresponding Smith Predictor architecture in line with Figure 10(c) is therefore as shown in

Figure 12. In this figure, IL is the inner loop transfer function and quantities with hats are

the modeled values of the actual physical parameters. The part shown in the dotted box is

the controller C2 defined in Figure 8. In can be seen that this part as a whole has two inputs

(the reference and the feedback) and a single output, the actuator command. For digital

implementation, the blocks in this part can be therefore composed into two transfer functions,

to avoid algebraic loops. These are then transformed to a discrete time transfer functions using

the bilinear transform, s = 2T

z−1z+1 [4]. As described above, if the models were exact, then the

Smith Predictor has the effect of moving the undesirable inner loop dynamics out of the outer

loop. If the model were not exact, it can be verified that the transfer function becomes

fachieved

fdesired

=1

1 + kLC[mst(1+IL)−mst(1+IL)]s2+[cst(1+IL)−cst(1+IL)]s+[kst(1+IL)−kst(1+IL)]

msts2+csts+kst

IL

As the stiffness of the added compliance decreases relative to the structure stiffness and the oil

column stiffness, the sensitivity of the performance of the Smith Predictor to modeling error

decreases. This is a further benefit of the added compliance.

7. EXPERIMENTAL RESULTS

Experiments were performed using a small-scale pilot test setup shown in Figure 13 to study

the performance of the proposed force control strategy, before it was applied to large-scale

actuators. A hydraulic actuator with 1 kN (2.2 kip) force capacity and 100 mm (4 in) stroke was

used. The actuator was fitted with a MTS 252.22 two-stage servo-valve with a 19 liters/minute

(5 gpm) flow capacity. The MTS FlexTest GT system was used for the inner loop controller.




Σ Σ ( )2 2

1

2st st st stm s sζ ω ω+ ++

+

+

-

Desired Force, f

Achieved Force

Added Compilance Structure

kLC1/kLC ILΣ+

+

2

ˆ

ˆ ˆˆ ˆLC

st st st LC

k

m s c s k k+ + +

1

ˆIL Σ+

-

error

Figure 12. Smith Predictor structure for displacement compensation

The outer loop controller was implemented using using Simulink and xPC Target [8]. For the

SDOF structure, a simple one story shear building model was used. A 305mm x 203mm x 25mm

(12in 8in x 1in) steel plate served as the floor, while four 12.7mm (0.5in) diameter aluminum

threaded rods served as columns. Braces were installed in the transverse direction on both

sides of the structure to limit out-of-plane motion. Lead blocks are used to provide additional

mass. Two different damping scenarios were considered for the structure — one with merely

the inherent damping in the structure, and another with model dashpots installed as shown

in Figure 13(a). Helical springs were used for the added compliance as show in Figure 13(c).

Four compression-only springs were used. They were pre-compressed so that they could act in

both tension and in compression. The properties of the structure and the added compliance

are summarized in Table I.

The inner loop controller C1 was tuned for position tracking, and the resulting frequency




mst kst ωst ζst kLC

Case 1 77.27kg (170lb) 16.67N/m (156lb/in) 3 Hz 0.01 12.57N/m (111lb/in)

Case 2 0.17

Table I. Structure Properties

Dashpot

Braces

Mass

(a) The SDOF structure

Load CellServovalve

Stroke

Hydraulic Supply

Reaction FrameStructure Displacement

Transducer

(b) The hydraulic actuator (c) Spring used for added compliance

Figure 13. Experimental Setup




0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Frequency (Hz)

x achi

eved

/xde

sire

d

(a) Magnitude

0 1 2 3 4 5 6 7 8 9 10−20

−18

−16

−14

−12

−10

−8

−6

−4

−2

0

Frequency (Hz)

Pha

se (

degr

ess)

(b) Phase

Figure 14. Inner loop Frequency Response Function

response function of the inner loop is shown in Figure 14. For the frequency range of interest,

it is seen that the inner loop dynamics can be modeled as a pure time-delay, τ , of 5.6 ms.

The force control performance was studied by measuring the frequency response of the ratio

of the applied force to the desired force. This was done using a crest factor-minimized multi-

sine input [13] for the desired force. Figure 15(a) shows the results for the structure with low




damping (ζst = 0.01). For the case with C2 = 1, the analytically obtained FRF considering

the actuator as a pure time-delay of 5.6 ms agrees well with the experimentally measured

FRF. This implies that that it is in fact reasonable to model the inner loop dynamics as

a pure delay. It is also seen that using a Smith predictor for C2 improves the force control

performance. However, the frequency response function still exhibits some drop (about 20 %) at

the resonant frequency of the structure and a bump at the resonant frequency of the structure

(about 10 %) with the added compliance. This is because the damping, being very small is

not known accurately and hence is modeled imprecisely. Figure 15(b) shows the results for

the structure with the added dashpots, and hence higher damping (ζst = 0.17). The frequency

response with C2 = 1 still shows a drop a the resonant frequency of the structure, but the

drop is smaller (about 20 %) because the zeros of the transfer function of equation (7) are

more highly damped. Since damping is modeled more accurately in the Smith Predictor, the

frequency response with compensation is almost ideally at one.

8. SUMMARY AND CONCLUDING REMARKS

From both the analytical and the experimental studies, the strategy of adding compliance and

providing displacement feedforward compensation appears adequate for dynamic force control

using hydraulic actuators over frequencies spanning resonances. The strategy does not use

force feedback, the measurement of which generally is noisier and is corrupted by stick-slip,

breakaway forces on seals, backlash etc. in the hydraulic actuator. The strategy results in two

controllers — an inner loop controller, a typical PID controller, whose role is to track a position

command, and outer loop controller whose role is to compensated for the inner loop dynamics.

The inner loop dynamics can be reasonable modeled as a pure time-delay. In this work, the




0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Frequency (Hz)

f achi

eved

/f desi

red

C2 = 1

Analytical with C2 = 1

C2 = Smith Predictor

(a) Case 1: ζst = 0.01

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Frequency (Hz)

f achi

eved

/f desi

red

Open Outer LoopC

2 = 1

C2 = Smith Predictor

(b) Case 2: ζst = 0.17

Figure 15. Force transfer function

outer loop controller has been designed using the Smith predictor approach. This requires

approximate models of the SDOF structure as well as of the inner loop dynamics. It is seen

that the performance of the system is less sensitive to the accuracy of these models when the

added compliance is made more flexible. The tuning of the inner loop controller also becomes

less sensitive to the dynamics of the SDOF structure with increase in this flexibility. The added




compliance thus provides an additional physical parameter in the dynamic force control system

design. The flexibility cannot be arbitrarily reduced, for this comes at the expense of increased

stroke requirement on the actuator. This method of force control has been successfully used

in a unified approach to real-time dynamic hybrid simulations [16]

ACKNOWLEDGEMENT

The authors gratefully acknowledge the financial support from the National Science Foundation

through the George E. Brown Network for Earthquake Engineering Simulation (NEES) development

program, grants #CMS-0086611 and #CMS-0086612.

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