Điều khiển kết cấu - Chương 7 pot

479
Chapter 7
Quasi-static Control Algorithms
7.1 Introduction to control algorithms
Referring back to Fig 6.1, an active structural control system has 3 main
components: i) a data acquisition system that collects observations on the
excitation and response, ii) a controller that identifies the state of the structure
and decides on a course of action and iii) a set of actuators that apply the actions
specified by the controller. The decision process utilizes both information about
how the structure responds to different inputs and optimization techniques to
arrive at an “optimal” course of action. When this decision process is based on a
specific procedure involving a set of prespecified operations, the process is said to
be algorithmic, and the procedure is called a “control algorithm”. A non-adaptive
control algorithm is time invariant, i.e., the procedure is not changed over the
time period during which the structure is being controlled. Adaptive control
algorithms have the ability to modify their decision making process over the time
period, and can deal more effectively with unanticipated loadings. They also can
upgrade their capabilities by incorporating a learning mechanism. This text is
concerned primarily with time invariant control algorithms which are well
established in the control literature. Adaptive control is an on-going research area
which holds considerable promise but is not well defined at this time. A brief
discussion is included here to provide an introduction to the topic.
The topic addressed in this chapter is quasi-static control, i.e., where the
480 Chapter 7: Quasi-static Control Algorithms
structural response to applied loading can be approximated as static response.
Since time dependent effects are neglected, stiffness is the only quantity available
for passive control. Active control combines stiffness with a set of pseudo-static
control forces. The quasi-static case is useful for introducing fundamental
concepts such as observability, controllability, and optimal control. Both
continuous and discrete physical systems are treated.
The next chapter considers time- invariant dynamic feedback control of

multi-degree-of freedom structural systems. A combination of stiffness, damping,
and time dependent forces is used for motion control of dynamic systems. The
state-space formulations of the governing equations for SDOF and MDOF
systems are used to discuss stability, controllability, and observability aspects of
dynamically controlled systems. Continuous and discrete forms of the linear
quadratic regulator (LQR) control algorithm are derived, and examples
illustrating their application to a set of shear beam type buildings are presented.
The effect of time delay in the stability of LQR control, and several other linear
control algorithms are also discussed.
7.2 Active prestressing of a simply supported beam
Passive prestressing
The concept of introducing an initial stress in a structure to offset the stress
produced by the design loading is known as prestressing. This strategy has been
used for over 60 years to improve the performance of concrete structures,
particularly beams. The approach is actually a form of quasi-static control, where
the variables being controlled are the stresses. Figure 7.1 illustrates prestressing of
a single span beam with a single cable. When the cable shape is parabolic, the
tension introduced in the cable creates an “upward” uniform loading, w
o
, that is
related to the tension by
(7.1)
The initial moment distribution is parabolic, and the moment is negative
according to the conventional notation.
w
o
L
2
8d
T=

7.2 Active Prestressing of a Simple Supported Beam 481
Fig. 7.1: Passive prestressing scheme
Suppose the design loading is a concentrated force that can act at any point
on the span. The maximum positive moment due to the force occurs when the
force acts at mid-span, and the resultant positive moment at mid-span is given by
(7.2)
The initial mid-span moment is negative and equal to
(7.3)
If the prestress level is selected such that
(7.4)
which requires
(7.5)
L
d
w
o
(-)
M+
w
o
L
2
/8
(+)
PL/4
M+
P
cable
d = distance
between top

and bottom
location of
the cable
M
L
2



PL
4

w
o
L
2
8
–=
M
L
2



w
o
L
2
8
–=

w
o
L
2
8

1
2

PL
4



M
*
==
T
M
*
d

T
*
≡=
482 Chapter 7: Quasi-static Control Algorithms
then the maximum positive and negative moments are equal. The cross section
can now be proportioned for , which is 1/2 the design moment corresponding
to the case of no prestress. This reduction is the optimal value; taking will
increase the initial moment beyond and result in the cross-section being

controlled by the initial prestress. The limitation of this approach is the need to
apply the total prestress loading prior to the application of the actual loading.
Since the tension is not adjusted while the loading is being applied, the scheme
can be viewed as a form of passive control. The best result that can be obtained
with prestressing for this example is equal design moment values for the
unloaded and loaded states.
Active prestressing
Suppose the cable tension can be adjusted at any time. The equivalent
uniform upward loading due to the cable action can now be considered to be an
active loading. Deforming as the equivalent active loading and noting eqn
(7.1), the loading is related to the “active” tension force, T(t ), by
(7.6)
The time history of T(t ) can be established using simple static equilibrium
relations. Given the spatial distribution and time history of the loading, T(t )is
determined such that the maximum moment at any time is less than the design
moment for the cross-section.
When the applied loading is uniformly distributed, the moment
distributions are similar in form. The expression for the net positive moment has
the form
(7.7)
Enforcing the constraint on the maximum moment, which occurs at mid-span,
(7.8)
results in the following control algorithm,
(7.9)
M
*
TT
*
>
M

*
w
a
w
a
8d
L
2



Tt()=
M
*
Mxt,() wt() w
a
t()–()
1
2

Lx x
2
–()=
wt() w
a
t()–()
L
2
8

M
*
≤
1. w
a
t() 0 for wt()
8
L
2

M* w
s
≡≤=
2. w
a
t() wt() w
s
for wt() w
s
>–=
7.2 Active Prestressing of a Simple Supported Beam 483
No action needs to be taken until reaches , since the maximum moment is
less than . Above this load level, the active loading counteracts the difference
between and . With active prestressing, the constraint imposed on the
initial prestressing is eliminated. Theoretically, the total applied load can be
carried by the active system for this example. This result is due to the fact that the
moment distributions for the actual and active loadings have the same form.
When these distributions are different, the effectiveness of active prestressing
depends on the difference between the distributions. The following discussion
addresses this point.

Consider the case where the loading is a concentrated force that can act at
any point on the span, and the prestressing action is provided by a single cable.
The moment diagrams for the individual loadings are shown in Fig 7.2. When
these distributions are combined, there is a local positive maximum at point B, the
point of application of the load, and possibly also at another point, say C.
Whether the second local negative maximum occurs depends on the level of
prestressing. As is increased, the positive moment at B decreases, and the
negative moment at C increases. For a given position of the loading, the control
problem involves establishing whether can be selected such that the
magnitudes of both local moment maxima are less than the prescribed target
design value, , indicated in Fig 7.2. With passive prestressing, the optimal
prestressing scheme produced a 50% reduction in the required design moment,
i.e., it resulted in =0.5(PL/4). Whether an additional reduction can be
achieved with active prestressing remains to be determined.
ww
s
M
*
wt() w
s
w
a
w
a
M
*
M
*
484 Chapter 7: Quasi-static Control Algorithms
Fig. 7.2: Active prestressing scheme for a concentrated load

The net moment is given by
Region A-B
(7.10)
(+)
P
P
ab
L

aLa–
w
a
L
2
8

–
(-)
M+
M+
parabola
low prestress
high prestress
M+
M
net
M
*
+
M

*
–
AB
C
D
x
c
x
Mxt,()
Px L a–()
L
w
a
Lx x
2
–
2



–=
7.2 Active Prestressing of a Simple Supported Beam 485
Region B-C-D
(7.11)
Specializing eqn (7.10) for leads to
(7.12)
The location of the second maxima is established by differentiating eqn (7.11) with
respect to x and setting the resulting expression equal to 0. This operation yields
(7.13)
The value for M at has the following form:

(7.14)
When , the maximum negative moment occurs outside the span, and
is taken as 0.
The control algorithm is established by requiring
(7.15)
for all . Starting at , no action is required as increases until the moment
due to the force P is equal to . The limiting value is denoted as , and
determined with
(7.16)
When is greater than , the maximum positive moment, , is set equal to
,
(7.17)
Solving eqn (7.17) for leads to
(7.18)
The last step involves checking whether for this value of , the maximum
negative moment, , exceeds .
Mxt,()
Pa L x–()
L
w
a
Lx x
2
–
2



–=
xa=

M
+
Mat,()≡ aL a–()
P
L

1
2

w
a
–


=
x
c
L
2

Pa
Lw
a
+=
xx
c
=
M
-
Mx

c
t,()≡ Lx
c
–()
Pa
L

w
a
x
c
2
– for x
c
L<=
x
c
L> M
–
M
+
M
*
≤
M
-
M
*
≤
aa0= a

M
*
a
s
P
L

a
s
La
s
–()M
*
=
aa
s
M
+
M
*
aL a–()
P
L

1
2

w
a
–



M
*
=
w
a
w
a
2
P
L

M
*
aL a–()
–


=
w
a
M
-
M
*
–
486 Chapter 7: Quasi-static Control Algorithms
It is convenient to work with dimensionless variables for x and M.
(7.19)

where
(7.20)
The factor, f, can be interpreted as the “reduction” due to prestressing. No
prestress corresponds to f=1; passive prestress for this loading and prestressing
scheme corresponds to f=0.5. Using this notation, is given by
(7.21)
The dimensionless form of eqn (7.18) is written as
(7.22)
Lastly, the dimensionless peak negative moment is expressed as
(7.23)
where
(7.24)
The peak negative moment is a function of the position coordinate, , and
the design moment reduction factor, f. Since must be less than 1 for all values
of between 0 and 0.5, the magnitude of f is constrained to be above a limiting
value, f
min
. Figure 7.3 shows plots of vs. for a range of values f. For this
case, the limiting value of f is equal to 0.26. Therefore, with active prestressing, the
design moment can be reduced to 50% of the corresponding value for passive
prestressing. The influence line for the cable tension required for the “optimal”
active prestressing algorithm is plotted in Fig 7.4. Also plotted is the required
tension corresponding to f=0.5, the optimal passive value. As expected, lowering
the cross-sectional design moment results in an increase in the required cable
tension. In order to arrive at an optimal design, the costs associated with the
material (cross-section) and prestressing need to be considered.
x
x
L
=

M
M
M
*
=
M
*
f
PL
4



=
a
s
a
s
1
2

11f–()
12⁄
–[]=
w
a
L
2
M
*

8d
M
*



T≡
8
f

2
a 1 a–()
– ga f,()==
M
-
M
*
M
-
1 x
c
–()
4a
f

gx
c
2
–==

x
c
1
2

4a
fg
+=
a
M
-
a
M
-
a
7.2 Active Prestressing of a Simple Supported Beam 487
Fig. 7.3: Influence lines for the peak negative moment
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
−1.8
−1.6
−1.4
−1.2
−1
−0.8
−0.6
−0.4
−0.2
0
M
–

aL⁄
f=0.2
f=0.3
f=0.4
f=0.5
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
−1.8
−1.6
−1.4
−1.2
−1
−0.8
−0.6
−0.4
−0.2
0
M
–
aL⁄
0.27
0.26
0.25
488 Chapter 7: Quasi-static Control Algorithms
Fig. 7.4: Influence lines for the optimal cable tension
Active prestressing with concentrated forces
In this section, the use of concentrated forces to generate prestress moment
fields is examined. The design loading is assumed to be a single concentrated
force that can act anywhere along the span.
Example 7.1: A single force actuator
Consider the structure shown in Fig (1). The active prestressing is provided

by a single force, F, acting at mid-span. This loading produces 2 local moment
maxima, M
1
and M
2
. The moment at mid-span may be negative for certain
combinations of and F, and therefore it is necessary to check both M
1
and M
2
when selecting a value for the control force. Adopting the strategy discussed
earlier, the control algorithm is based on the following requirements
(1)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0
1
2
3
4
5
6
aL⁄
8d
PL 4⁄



T
f=0.26
f=0.5

a
M
1
M
*
≤
M
2
M
*
≤
7.2 Active Prestressing of a Simple Supported Beam 489
for all where
(2)
(3)
and is the design moment for the cross-section.
Figure 1
a
M
1
Pa L a–()
L

Fa
2
–=
M
2
Pa
2

FL
4
–=
M
*
(+)
P
P
aL a–()
L

aLa–
FL
4

–
(-)
M+
M+
L/2
F
M+
M
1
M
2
490 Chapter 7: Quasi-static Control Algorithms
Shifting to dimensionless variables,
(4)

transforms the equations to
(5)
(6)
No action is required for where is defined by eqn (7.21).
(7)
When , the force is selected such that .
(8)
The corresponding expression for is
(9)
Figures 2 and 3 show the variation of and with and f. The limiting value
of f for active prestressing is 0.345; when f>0.345, the negative moment at mid-
span is greater than the design moment, . For passive prestressing, the
optimal solution is f=0.5. Shifting from passive to active control results in an
additional 30 percent reduction in the allowable design moment.
a
a
L
=
M
*
f
PL
4

=
M
i
M
i
M

*
=
F
F
P
=
M
1
4
f

a 1 a–()
aF
2
–=
M
2
1
f

2aF–[]=
aa
s
≤ a
s
a
s
1 a
s
–()

4
f

1≡
aa
s
> M
1
1≡
F 21 a–
f
4a
–


=
M
2
M
2
4a 2–
f

1
2a
+=
M
2
F a
M

*
7.2 Active Prestressing of a Simple Supported Beam 491
Figure 2
Figure 3
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
−4
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
0.2
0.25
0.3
0.35
0.4
0.45
0.5
M
2
a
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0
0.2
0.4

0.6
0.8
1
1.2
1.4
0.2
0.25
0.3
0.35
0.4
0.45
0.5
F
a
492 Chapter 7: Quasi-static Control Algorithms
Example 7.2: Two force actuators
The previous example showed that the effectiveness of a prestressing
scheme depends on the difference between the moment distributions for the
applied loading and the prestressing forces. In the case of a single control force
applied at mid-span, the limiting condition occurs when the applied load is near
the end support, where the difference in the moment distributions is a maximum.
Increasing the number of control forces provides the capability to modify the
“shape” of the “prestress” moment distribution to conform better with the
applied moment field, and therefore increase the amount of prestressing that can
be applied. This example illustrates the use of self-equilibrating control force
systems which provide the maximum flexibility for adjusting the moment field.
Consider the self-equilibrating force system shown in Fig. 1. This force
system produces a bilinear moment field which is local, i.e., confined to the
loaded region. Therefore, perturbing the control force magnitude, F, has no effect
outside of this region.

Figure 1
Applying a set of these self-equilibrating systems results in a piecewize
linear moment distribution. Figure 2 illustrates the case of 2 force systems located
immediately adjacent to each other. The corresponding moment field is defined in
terms of 2 force parameters, F
1
and F
2
.
M+
F F/2
F/2
l
l
Fl
2

–
7.2 Active Prestressing of a Simple Supported Beam 493
Region A-B
(1)
Region B-C
(2)
Region C-D
(3)
where
(4)
Figure 2
Example 7.1 treated the case of a single actuator deployed on a simply
supported beam subjected to a single concentrated force that can act at any point

on the beam. Suppose the control force system now consists of 2 local moment
fields centered at the third points of the span. Figure 3 shows the 2 loading
scenarios for this example. The moments at B,C, and D corresponding to the
different loading scenarios are:
M
M
1
l

– xx
A
–()=
M
M
1
l

lx
B
x–+()–
M
2
l

xx
B
–()–=
M
M
2

l

lx
c
x–+()–=
M
1
F
1
l
2
= M
2
F
2
l
2
=
M+
F
1
2

AB C D
x
F
1
F
2
2

F
1
2

F
2
F
2
2

l
l
l
(-)
M
1
M
2
494 Chapter 7: Quasi-static Control Algorithms
Region A-B (Fig 3a):
(5)
(6)
(7)
Region B-C (Fig 3b):
(8)
(9)
(10)
Figure 3a
M

p
PL a 1 a–()()3aM
1
–=
M
B
2
3

PLa M
1
–=
M
C
1
3

PLa M
2
–=
M
p
PL a 1 a–()()M
1
23a–()– M
2
3a 1–()–=
M
B
1

3

PL 1 a–()M
1
–=
M
C
1
3

PLa M
2
–=
M+
L/3 L/3
L/3
A
B
C
D
P
aaL=
(+)
M
1
M
2
(-)
M
P

M
B
M
C
7.2 Active Prestressing of a Simple Supported Beam 495
Figure 3b
Let represent the design moment. Expressing as a fraction of the
maximum moment for the case where there is no prestressing,
(11)
and working with dimensionless moments,
(12)
transforms eqns (5) thru (10) to the following:
Region A-B
(13)
(14)
(15)
M+
L/3 L/3
L/3
A
B
C
D
P
aaL=
(+)
M
1
M
2

(-)
M
P
M
B
M
C
M
*
M
*
M
*
f
PL
4

=
M
()
M
()
M
*
=
M
p
4
f

a 1 a–()()3aM
1
–=
M
B
8
3 f

aM
1
–=
M
C
4
3 f

aM
2
–=
496 Chapter 7: Quasi-static Control Algorithms
Region B-C
(16)
(17)
(18)
The control objective for this example is to limit the peak value of each of
the dimensionless moment variables to be less than unity.
(19)
Since there are 3 constraints and only 2 control parameters, the problem is
overconstrained. The strategy followed here is based on determining and
using 2 of the constraints in eqn (19), and then adjusting f such that the third

constraint is also satisfied. Figure 4 shows the variation in the moment measures
with , the coordinate defining the position of the load, corresponding to the
following choice of constraints:
Region A-B
(20)
Region B-C
(21)
The minimum value of f is equal to 0.228. This value is controlled by the
constraint on in the region A-B. For the case of a single actuator applied at
M
p
4
f

a 1 a–()M
1
23a–()– M
2
3a 1–()–=
M
B
4
3 f

1 a–()M
1
–=
M
C
4

3 f

aM
2
–=
M
P
1≤ M
B
1≤ M
C
1≤ 0 a
1
2

≤≤
M
1
M
2
a
M
P
+1=
M
C
+1=
M
B
1<

M
P
+1=
M
C
+1 4 3a 1–()–=
M
B
1<
M
B
7.2 Active Prestressing of a Simple Supported Beam 497
mid-span, the optimum value for f was found to be 0.345. Applying 2 force
actuators leads to an additional reduction in the “permissible” design moment.
Figure 4a
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
M
1
a
f 0.228=
498 Chapter 7: Quasi-static Control Algorithms

Figure 4b
Figure 4c
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
M
2
a
f 0.228=
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
−1.5
−1
−0.5
0
0.5
1
M
B
a
f 0.228=
7.2 Active Prestressing of a Simple Supported Beam 499
Figure 4d
A general active prestressing methodology

The discussion to this point has been concerned with a specific loading and
a specific prestressing scheme. In what follows, a general methodology for
dealing with the combination of an arbitrary design loading and prestressing
scheme applied to a simply supported beam is described, and the control
algorithm corresponding to a particular choice of error measure is formulated.
This methodology is also applicable for displacement control which is discussed
in the following section.
Let denote the moment due to the design loading, the
moment generated by the prestress system, and M(x) the net moment. By
definition,
(7.25)
The design objective is to limit the magnitude of to be less than , the
design moment for the cross-section.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
M
C
a
f 0.228=
M

d
x() M
c
x()
Mx() M
d
x() M
c
x()+=
Mx() M
*
500 Chapter 7: Quasi-static Control Algorithms
(7.26)
Equation (7.26) imposes a constraint on the magnitude of .
(7.27)
These limits establish the lower and upper bounds for . Given , one
generates the limiting boundaries and then decides on a “target” distribution for
.
Fig. 7.5: Limiting prestress moment fields
Figure (7.5) illustrates the process of establishing the desired distribution
for the control moment. The curves shown in Fig (7.5b) correspond to
; allowable values of are defined by the shaded area. The
distribution corresponding to selecting the minimum allowable value of
at each x is plotted in Fig (7.5c). Assuming magnitude is the dominant measure,
M
d
M
c
+ M
*

≤ 0 xL≤≤
M
c
M
d
– M
*
M
c
M
*
M
d
–≤≤– 0 xL≤≤
M
c
x() M
d
x()
M
c
x()
1
0
-1
-2
a
b
c
d

e
(a)
(b)
0
1
3
-1
-2
1
0
-1
-2
-4
a
b
c
d
e
(c)
upper bound
lower bound
M
d
M
*

M
c
M
*

M
c
*
M
*

Permissible region
Optimum solution
M
c
M
d
– M
*
±= M
c
M
c
x()
7.2 Active Prestressing of a Simple Supported Beam 501
this distribution represents the optimal “target” prestress moment field.
Let denote the desired “target” prestress moment field. Suppose
the actual prestress moment field is a linear combination of r individual fields,
(7.28)
where are moment amplitude parameters and are dimensionless
functions. The error associated with a specific choice of moment parameters is
represented by the difference, ,
(7.29)
Ideally, one wants for . However this goal cannot be achieved

when is an arbitrary function, and it is necessary to work with an
approximate error condition established using collocation, the least square
method, or some other weighted residual scheme.
The least square method is based on taking the integral of as a
measure of the accuracy of the approximation represented by eqn (7.28). This
integral is denoted as J.
(7.30)
In general, J is a function of the r moment parameters. Equations for these
parameters are generated by requiring J to be stationary.
(7.31)
Expanding eqn (7.31) results in the following linear matrix equation,
(7.32)
where are r’th order matrices, and the elements of and are:
(7.33)
(7.34)
Given ,one determines and then solves for . This solution produces
the least value for J, for a particular set of ‘s. A sense of convergence can be
M
c
*
x()
M
c
x() m
j
ψ
j
x()
j 1=
r

∑
=
m
j
ψ
j
x()
ex()
ex() M
c
*
x() M
c
x()–=
ex() 0=
0 xL≤≤
M
c
*
ex()()
2
J
1
2

e
2
xd
0
L

∫
=
m
i
∂
∂J
0= i 12… r,, ,=
am b=
abm,,
a
b
a
ij
Ψ
i
x()Ψ
j
x()xd
0
L
∫
=
ij, 12… r,,=
b
i
Ψ
i
x()M
c
*

x()xd
0
L
∫
=
M
c
*
x() bm
Ψ
502 Chapter 7: Quasi-static Control Algorithms
obtained by expanding the set of basis functions, and comparing the
corresponding values of J. It should be noted that the exact condition, e(x)=0 for
0<x<1, is generally not satisfied by eqn (7.32).
This formulation works with continuous functions, and requires the
evaluation of a set of integrals. It is more convenient to work with vectors rather
than functions, since the computation reduces to matrix operations. Suppose the
moment is monitored at n points within the interval . The desired
prestress moment vector is of order n .
(7.35)
Evaluating eqn (7.28) at these observation points leads to
(7.36)
where is of order . The error vector is taken to be the difference between
and ,
(7.37)
When and the individual prestress moment fields are linearly
independent, is non-singular and it is possible to determine an that exactly
satisfies .
(7.38)
When , a least square procedure can be used to establish an approximate

solution for . The error measure is taken as the norm of .
(7.39)
Requiring J to be stationary with respect to leads to an equation having the
same form as eqn (7.32), with and now given by
(7.40)
(7.41)
The following example illustrates the application of the discrete formulation.
0 xL≤≤
1×
M
c
*
M
c 1,
*
M
c 2,
*
… M
cn,
*
,,,{}=
M
c
Ψm=
Ψ nr×
M
c
*
M

c
eM
c
*
M
c
– M
c
*
Ψm–==
rn=
Ψ m
e 0=
m Ψ
1–
M
c
*
=
rn≠
e 0= e
J
1
2

e
T
e J m()==
m
ab

a Ψ
T
Ψ=
b Ψ
T
M
c
*
=
7.2 Active Prestressing of a Simple Supported Beam 503
Example 7.3: Multiple actuators
Consider the design moment field shown in Fig (1). The longitudinal axis
is discretized with 10 equal segments, resulting in 11 (n=11) observation points.
Applying the criteria defined by eqn (7.27), and taking leads to the
bounding curves for plotted in Fig (2). The problem now consists of
generating a prestress moment distribution which lies between these bounds.
Suppose 4 self-equilibrating force system (r=4) that produce bilinear
moment fields are applied at equally spaced interior points. The corresponding
functions are shown in Fig (3a) and the typical field is plotted in Fig (3b). Since
the prestress moment field is defined in terms of the moments at only 4 fixed
points (3,5,7,9), and there are 9 interior points, the solution for the case of an
arbitrary target distribution will be approximate. Various solutions for a
particular target distribution are plotted in Fig (4). Curve (1) corresponds to
taking , , and . Curve (2) is the least square
solution for the 4 actuator system defined in Fig (3a). Curve (3) is based on the 7
actuator system defined in Fig (3c). The 3 additional actuators applied at points
2,4, and 6 eliminate the error up to point 6. Incorporating 2 more actuators at
points 8 and 10 would completely eliminate the error associated with the 4
actuator system. It produces a moment field that is fully contained within the
allowable zone and has the lowest cost as measured by the sum of the actuator

moment magnitudes, .
Figure 1: Moment due to design loading
M
*
1=
M
c
*
Ψ
m
1
m
2
1–== m
3
0= m
4
1=
Σm
i
M
d
3
2
1
-1
-2
1
2
3

45
6
78
9
10 11