Dynamic Force Control with Hydraulic Actuators Using Added...
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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
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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.
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4 M. V. SIVASELVAN ET. AL.
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
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DYNAMIC FORCE CONTROL 5
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.
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6 M. V. SIVASELVAN ET. AL.
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].
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DYNAMIC FORCE CONTROL 7
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
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8 M. V. SIVASELVAN ET. AL.
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.
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DYNAMIC FORCE CONTROL 9
Flexible Structure
Position Command
Added Compliance, kLC
Measured Force, f
Actuator in Closed-loop
Position Control
1 / kLC
Structure Model
+
+
+
+
Desired Force, f
(a) Using a model to obtain the structure displacement
Flexible Structure
Position Command
Added Compliance, kLC
Measured Force, f
Actuator in Closed-loop
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.
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10 M. V. SIVASELVAN ET. AL.
Σ Σ
( )2 2
1
2 st st stM s sζ ω ω+ +
+
+
+-
-
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
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DYNAMIC FORCE CONTROL 11
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
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12 M. V. SIVASELVAN ET. AL.
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.
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DYNAMIC FORCE CONTROL 13
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.
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14 M. V. SIVASELVAN ET. AL.
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
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DYNAMIC FORCE CONTROL 15
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
2 st st stM s sζ ω ω+ +
+
+
+
-
Desired Force, f
Achieved Force
Added Compilance
Structure
kLC1/kLC
xst
stLC
fx
k+
1
Figure 11. Desired feedback structure for displacement compensation
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16 M. V. SIVASELVAN ET. AL.
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.
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DYNAMIC FORCE CONTROL 17
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
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18 M. V. SIVASELVAN ET. AL.
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
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DYNAMIC FORCE CONTROL 19
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
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20 M. V. SIVASELVAN ET. AL.
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
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DYNAMIC FORCE CONTROL 21
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
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22 M. V. SIVASELVAN ET. AL.
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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