Harmonic Distortion from Variable Frequency Drives Harmonic ...
1- Independent Harmonic Control for Structural Engineering
Transcript of 1- Independent Harmonic Control for Structural Engineering
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Journal of Earthquake Engineering, Vol. 6, No. 3 (2002) 297314c Imperial College Press
INDEPENDENT HARMONIC CONTROL FOR
STRUCTURAL ENGINEERING
ALESSANDRO BARATTA and OTTAVIA CORBI
Department of Scienza delle Costruzioni,
University of Naples, via Claudio 21 Naples, 80125, Italy
Received 17 May 2001Revised 31 October 2001
Accepted 14 November 2001
In a previous paper [Baratta and Corbi, 1999] one has defined a procedure allowing toidentify a closed-loop control algorithm with feedback based on the whole record of theresponse time-history rather than on instantaneous response parameters. The controlforce results from control of each harmonic component of the forcing function, simplyintegrated over the frequency domain. Every harmonic is controlled, independently ofeach other, by a classical linear control whose coefficients are calibrated in way to makethe relevant response component a minimum compatibly with the control effort one
wants to apply at the corresponding frequency. The distribution of this control inten-sity over the frequency range remains a arbitrary choice; such a choice however lendsitself to be effectively assisted by intuition, much more than similar choices in otherprocedures (e.g.: the coefficients of the quadratic norms in the J-index optimization).The result is that every harmonic remains controlled by a different couple of optimalcoefficients (corresponding to the proportional and to the derivative terms in the linearcontrol law), and the overall control force for an arbitrary disturbance, after Fourierinverse transformation, is produced by feedback integration over the whole responsetime-history.
The procedure, tested with reference to simple and composed harmonic excitationsincoming a s.d.o.f. structural system, has proved a good agreement of the numerical re-sults with the theoretical treatment; furthermore it has shown that the main limit of such
an approach consists of referring the dynamic equilibrium solution to a particular solu-tion, that, neglecting the initial conditions, may introduce some unstable componentsin the oscillation. In the paper the effects induced in the controlled structural systemresponse by the adoption of the proposed procedure are deepened and an improved strat-egy is presented, able to overcome the detrimental transient effects determined by theoriginal algorithm. The final adopted control law is shown to achieve an improved timeresponse, both in the transient and in the steady-state field, in comparison to a controlstrategy based on classical linear control minimizing the response norm conditioned bya bounded control.
Keywords: Harmonic control; frequential decomposition; norm algorithm.
1. Introduction
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298 A. Baratta & O. Corbi
interest on this argument is proven by the large literature that is appearing on
many journals and by the many conferences that are holding all over the world. A
very important state of the art paper has been redacted by a number of promi-nent scientists in the field, and it documents the encouraging results that have been
obtained and that are currently being produced [Housner et al., 1996].
At least four possible approaches to Structural Control can be distinguished:
passive, active, semi-active and hybrid, each possessing its own characteristics and
requiring different hardware and software equipment [Housner and Masri, 1996;
Kobori, 1998]. In Active Control (AC) technology [Soong, 1990; Soong et al., 1994],
the control action is added by external forces generated by contrast with oscillating
masses (Active Mass Driver) or by contrast with an oscillating system that can be
damaged and possibly replaced after the quake (Inter-Active Control). Such forcesare driven during the motion by a CPU central on the basis of the instantaneous
sensing of the response parameters and according to a pre-established rule, thus
conferring a kind of artificial intelligence to the system. The rule governing the
control force can be set on the basis of a mathematical algorithm or on the basis of
learning devices such as neural networks or genetic algorithms, all of them aiming at
achieving an objective response that is considered optimal in some sense, essentially
to yield the maximum possible mitigation of the structures strength, at the expense
of the minimum engagement of the control force.
Particular AC systems make recourse to linear control laws (Linear Control)whose objectives are identified in some well defined functionals of the response and
of the control force (Quadratic Performance Index, Norm Response Functionals,
Instantaneous Optimal Control IOC, etc.) [Soong, 1990]. A different control
approach expressly studied to save the energy employed in the control process is the
Bounded State Control, that finds application in a class of algorithms designed to
reduce the controlled response variables (expressed in the state space) to predefined
bounds: a suitably chosen control force activates only when an overcoming of the
predefined threshold of any of the state variables is detected and lasts just for the
length of time necessary to reduce the variable to the allowed magnitude range.In previous papers the authors [Baratta and Corbi, 1999] have proposed a pro-
cedure finalised to define a linear control feedback algorithm for structural systems,
able to effectively counteract the incoming forcing function that, as well known,
can be viewed as the sum of a number of harmonic components. The procedure is
based on the superposition of control laws independently controlling every single
harmonic component of the external disturbance. The method allows a wide free-
dom in the distribution of the control effort over the frequency range, that can be
decided on the basis of some expectation concerning the power spectrum of the
excitation. Actually the algorithm, designed in the frequency field by developing aprocedure referred just to the steady-state domain, is able to achieve really satis-
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Independent Harmonic Control for Structural Engineering 299
effectiveness of the proposed procedure is widely investigated by simulation and the
chance to attenuate the influence of the transient motion by coupling the harmonic
control with a classical linear algorithm is tested.
2. The Harmonic Control Algorithm
2.1. Constrained optimal linear control for a harmonic excitation
Let consider a controlled s.d.o.f. structural system
mu(, t) + cu(, t) + ku(, t) + w(, t) = f(t) , (1)
subject to an harmonic forcing function f(t) = fo()e
jt
and a linear control laww(, t) = wu(, t) + wu(, t) = p()u(, t) + q()u(, t) (2)
being p() and q() the control parameters; a particular solution u(, t) of Eq. (1),
should satisfy
mu(, t) + cu(, t) + ku(, t) +p()u(, t) + q()u(, t) = f(t) (3)
and it can be set in the form
u(, t) = uo()ejt , u(, t) = juo()e
jt , u(, t) = 2uo()ejt ; (4)whence one deduces that uo() satisfies
uo() = H()fo() (5)
with
H() =1
[k +p() m2] +j[c + q()]
=k +p() m2 j[c + q()]
[k +p() m2]2 + 2[c + q()]2
|H()| = 1[k +p() m2]2 + 2[c + q()]2 .
(6)
Note that, assuming c > 0, |H()| is bounded for = 0, for any value of p() andfor any q() > c; it follows that the control Eq. (2) cannot introduce instability inthe solution u(, t). This does not mean that divergent motion cannot be triggered,
as it will be discussed in Sec. 3.1.
Note that, in the following, for simplicity of notation, the symbol x2 will be used
signifying |x|2 for any complex variable x. The control parameters p() and q()must be selected in way to be compatible with some design and functional issues.
A possible approach in Structural Engineering problems is to aim at the maximum
attenuation of the structure response by keeping the energy supply below a given
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300 A. Baratta & O. Corbi
w(|p,q) = [p() +jq()]H()fo() ;
w2(|p,q) = |p2() + 2q2()|H2()f2o ()(8)
the search for the control parameters, optimal for an harmonic excitation, can be
set in the form of the following problem of constrained minimum
minp,q
u2o(|p,q)f2o ()
= minp,q
H2(|p,q)
subw2(|p,q)
f2o ()= [p2() + 2q2()]H2(|p,q) C2o ()
(9)
where Co() is, for any given , a number less than unity, in that, as a rule, the
control force is generally required to be instantaneously smaller in magnitude than
the active force. With the position
G2(|p,q) = [k +p() m2] + 2[c + q()]2
V2(|p,q) = [p2() + 2q2()](10)
problem (9) can be written in the equivalent inverse formulation
maxp,q G
2(|p,q)sub G2(|p,q) C2Io()V2(|p,q) 0
(11)
with CIo = 1/Co. In order to solve the constrained extremum problem, let consider
the KuhnTuckers conditions, with the Lagrangian multiplier
(1 + )G2(|p,q)
p C2Io()
V2(|p,q)p
= 0
(1 + )G2(
|p,q)
q C2Io()V2(
|p,q)
q = 0
[G2(|p,q) C2Io()V2(|p,q)] = 0
0
(12)
yielding
(1 + )[k +p() m2] p()C2Io() = 0
(1 + )2[c + q()] C2Io()2q() = 0(13)
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Independent Harmonic Control for Structural Engineering 301
The case = 0 shall be discarded, in that it leads to the absolute minimum of the
objective function, i.e. G2(
|p,q) = 0. Assuming > 0 Eqs. (13) yield
p() = (k m2)(1 + )
(1 + C2Io())
q() = c(1 + )(1 + C2Io())
G2(|p,q) C2Io()V2(|p,q) = 0
> 0 .
(14)
By solving the third of (14), one gets the roots
1 =1 + CIo()
C2Io() 1, 2 =
1 CIo()C2Io() 1
. (15)
Since C2Io() > 1 and the Lagrangian multiplier must be positive, the second root
is definitely discarded and the first two of (14), with = 1, yield the control
parameters that comply with the optimal criterion expressed by (9)
p() =k m2
CIo()
1
; q() =c
CIo()
1
. (16)
Assuming that c > 0, the coefficient q() is always positive, while p() is negative
for values of larger than 2o =
k/m.
Note that for = o, p() = 0 and at this pulsation the control is exclusively
demanded to the derivative term. This result agrees with the fact that the deriva-
tive linear term turns out to be prominent in any control algorithm, especially at
resonance [Baratta and Di Paola, 1996]. Note anyway that this is not a choice, but
it is the optimal solution of the conditioned optimization problem in Eq. (9).
Note that if Co() is introduced as a even function of , also 1 is a even
function, and both the control coefficientsp() and q() are real and even functionsof .
2.2. The optimal harmonic control for a generic forcing function
Consider the structure subject to any generic forcing function f(t), whose Fourier
transform fo() exists
fo() =12
+
f(t)ejtdt ;
f(t) =12
+fo()e
jtd .(17)
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302 A. Baratta & O. Corbi
one gets
m 12
+
2uo()ejtd + c 12
+
juo()ejtd
+ k12
+
uo()ejtd +
12
+
p()uo()ejtd
+12
+
jq()uo()ejtd =
12
+
fo()ejtd (18)
whence one concludes that
u(t) = 12
+
uo()ejtd = 1
2
+
u(, t)d (19)
with
u(t) =12
+
u(, t)d = j12
+
uo()ejtd
u(t) =12
+
u(, t)d = 12
+
2uo()ejtd
(20)
is a particular solution of the equation
mu(t) + cu(t) + ku(t) + w(t) = f(t) (21)
yielding the response to the excitation f(t) of the structure, controlled by the active
force
w(t) =12
+
p()uo()ejtd +
12
+
jq()uo()ejtd
=1
2+
p()u(, t)d +1
2+
q()u(, t)d = wu(t) + wu(t) .
(22)
For a generic bounded-support forcing function (like in case of seismic action) the
existence of all the above transforms is ensured, and uo() and u(t) constitute a
Fourier transform pair
uo() =12
+
u()ejd; juo() =12
+
u()ejd . (23)
It is possible to manipulate the algorithm yielding the control force in Eq. (22).
In fact, considering the inverse Fourier transforms of p() and q()
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Independent Harmonic Control for Structural Engineering 303
introducing Eqs. (23) into (22) and distinguishing the two terms wu(t) and wu one
can write
wu(t) =12
+
p()uo()ejtd =
1
2
+
p()
+
u()ejd
ejtd
=12
+
12
+
p()ej(t)d
u()d
=12
+
P(t )u()d (25)
wu(t) = 12
+
jq()uo()ejtd
=1
2
+
q()
+
u()ejd
ejtd
=12
+
12
+
q()ej(t)d
u()d
=1
2 +
Q(t
)u()d . (26)
Note that P(x) and Q(x) are real and even functions of x, in that both p() and
q() are real and even functions of .
One should remark that the response u(t) [Eq. (19)] results from the superposi-
tion of harmonic components, each optimised in the sense of the problem (9), with
the relevant component of the control obeying the constraint inequality. Hence the
final control force remains bounded by
|w(t)| =1
2 +
w(|p,q)e
jt
d 12
+
Co()|fo()|d . (27)
Whence one understands how the function Co() can be properly designed in such
a way to have the desired threshold level of the control force.
With reference to the practical implementation of the proposed control
algorithm, note that any control rule requires some prediction; this can, then, be
reviewed as a common problem, met even in the optimal calibration of control coef-
ficients in classical linear control that is nevertheless widely applied (see i.e [Soong,
1990; Chap. 3. Baratta and Corbi, 2000]). So, practical feedback implementation
will be forced to accept some manipulation and/or approximation. The problemis under current development and it is not yet fully afforded in this paper, which
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304 A. Baratta & O. Corbi
2.3. Purely real forcing function
In most cases (i.e. when a structure is invested by a strong earthquake) the dis-turbance f(t) is a purely real function, and it starts finding the structure in some
known status at time t = 0. The structure response on the semi-axis t > 0 can
be made independent on the previous time-history, by setting proper conditions at
t = 0. Hence one can consider the function f(t) to be a real even function of t
defined on the whole real axis. In this case fo() turns out to be as well a real even
function of: the real uoR() and the imaginary uoI() parts ofuo() are an even
and an odd function of respectively
uoR() =
k +p()
m2
[k +p() m2]2 + 2[c + q()]2 fo()uoI() = [c + q()]
[k + p() m2]2 + 2[c + q()]2 fo()(28)
so that also u(t), as expressed by Eq. (19), is real (but it is not even, in general).
On such an understanding, one can directly refer to the one-sided transforms, so
that all above formulas can be written in the form
fo() = 2
+
0
f(t)cos tdt ;
f(t) =
2
+0
fo()cos td
(29)
u(t) =
2
+0
[uoR()cos t uoI()sin t]d (30)
u(t) = j1
2 +
uo()ejtd
=12
+
[juoR() uoI()] [cos t +j sin t]d
=
2
+0
[uoR()sin t + uoI()cos t]d (31)
u(t) = 12
+
2uo()ejtd
= 12
+
2[uoR() +juoI()] [cos t +j sin t]d
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Independent Harmonic Control for Structural Engineering 305
wu(t) =12
+
p()uo()ejtd
=
2
+
p()[uoR()cos t uoI()sin t]d
wu(t) =12
+
jq()uo()ejtd
=
2
+0
q()[uoR()sin t + uoI()cos t]d (33)
P(x) =
2
+
0
p()cos xd ; Q(x) =
2
+
0
q()cos xd (34)
wu(t) =
2
+0
P(t )u()d ; wu(t) =
2
+0
Q(t )u()d (35)
Since the initial conditions at t = 0 for u(, t) are
u(, 0) = uo() =
[k +p() m2]
G2(|p,q) j[c + q()]
G2(|p,q)
fo()
u(, 0) = juo() =
2[c + q()]G2(|p,q) +j
[k +p() m2]G2(|p,q)
fo()
(36)
the particular solution u(t) obeys the following initial conditions
u(0) =
2
+0
uoR()d =
2
+0
[k +p() m2]G2(|p,q) fo()d
u(0) =
2
+0
uoI()d =
2
+0
2[c + q()]
G2(|p,q) fo()d(37)
3. Considerations on the Initial Conditions Effect
3.1. The role of initial conditions
By looking at the results of the optimisation given by (16) and depicted in Fig. 1,
where the objective function and the constraint condition are displayed through
their level curves, it is possible to observe that in many cases the control coefficient
p() is negative; it follows that the controlled response for a single harmonic force
component [Eq. (3)] is less robust than the original one, or even that it becomes
unstable (this is the case if p() < k).This happens although |H()| in Eq. (6) is smaller than the modulus of the
f f ti f th t ll d t t |H ( )| I th d
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306 A. Baratta & O. Corbi
100
p10008006004002000
q200
150
50
0
Co
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Independent Harmonic Control for Structural Engineering 307
can be unstable or, at least, trigger somewhat different responses with varying the
initial conditions.
In order to illustrate this effect, consider the structure acted on by the harmonicforce
f(t) = a1 sin t + a2 cos t . (38)
The optimally controlled (in the sense of Sec. 2.1) structure response obeys the
equation
mu(, t) + c()u(, t) + k()u(, t) = f(t) (39)
where
k() = k +p() ; c() = c + q() . (40)
The particular solution in the form equivalent to (4) is
u(, t) = u1()sin t + u2()cos t
u(, t) = [u1()cos t u2()sin t](41)
with
u1() =[k() m2]a1 + c()a2[k
() m2
]2
+ c
()2
2
;
u2() =[k() m2]a2 c()a1[k() m2]2 + c()22 .
(42)
Equation (41) will be referred to as the stationary solution; it obeys the following
initial conditions at t = 0
u(, 0) = u2()
u(, 0) = u1() .(43)
Equation (42) proves that the stationary solution remains stable provided thatboth k and c in the original equation are larger than zero, as can be easily verified
considering that p(0) > 0, q() > 0 [see (16)].By contrast, let consider the general integral of Eq. (39). The characteristic
equation is
m2 + c() + k() = 0 (44)
whose roots are
1 =c() + c()2 4k()m
2m; 2 =
c() c()2 4k()m2m
(45)
and the general integral of Eq. (39) is
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308 A. Baratta & O. Corbi
If k() 0 it results 1 0 and the general integral is unbounded unless A = 0.This means that in this case only a one-parameter class of bounded solutions exists,
corresponding to initial conditions
U(0|B) = B + u2 ; U(0|B) = B + u1 any B (47)If k() > 0 but k()m c2 it results 1 0 and 2 < 0. If k()m > c2, 1and 2 are complex numbers and the free component of the motion, as well known,
has a decaying oscillatory feature. In both cases the general integral is bounded.
Nevertheless, the motion can be very heavily affected by initial conditions other
than the stationary ones in (43).
If, for instance, one searches for the solution verifying homogeneous initial con-
ditions, as is the case if it is necessary to study the effect of a transient disturbanceon a structure, one has
A =u1 2u2
2 1 ; B =1u2 u1
2 1 . (48)
Some plots of the structure response vs. time are quoted in Figs. 24, corresponding
to the following values of the structure and force parameters
k = 900 kg/cm; m = 1 kgs2/cm; c = 1.5 kgs/cmo = 30 s1 ; = 5% ; To
0.21 s ;
= 2o = 60 s1; a1 = a2 = 1 kg .
(49)
In (49), o and To are the own pulsation and the own vibration period of the
structure, and is the damping coefficient. From Fig. 2, wherep() = 900.3 kg/cmand q() = 0.5 kgs/cm are assumed, one can notice that, even if the systembecomes unstable (k() < 0), the stationary response remains bounded while,
by contrast, the homogeneous solution diverges. In the limit case when k() = 0
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Independent Harmonic Control for Structural Engineering 309
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(p() = k and q() = 0.5 kgs/cm are now considered) the homogeneous responseis bounded but it takes values much higher than the stationary one [Fig. 3], while
Fig. 4 (where one assumes p() = 675 kg/cm and q() = 0.375 kgs/cm) provesthat, even if the system remains stable (k() > 0), the homogeneous response can
be very different from the stationary one, for a rather long transient time ( t 8To).If one looks at the control force, one realizes that in the transient phase a surplus
control may be required, disturbing the optimal character of the algorithm.
3.2. Attenuation of the initial conditions effect
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310 A. Baratta & O. Corbi
proposed control algorithm, consisting of the addition of a free- oscillation compo-
nent to the stationary solution. One can believe that it is possible to attenuate such
disturbance by adding a further control specifically designed for mitigation of freeoscillations. It is well known [Soong, 1990; Baratta and Corbi, 2000] that a classical
linear control law is optimal to this purpose; the only problem consists of suitably
calibrating the relevant coefficients.
This can be done following the procedure set forth in [Baratta et al., 1998] under
the name norm-control, based on the constrained minimization of the energy of the
impulse response function of the controlled structure
hc(x) =1
cdecx sin cdx ; c =
c + c
2m
; cd = k + km
2c (50)
with k and c the coefficients of a classical linear control law
w(t) = ku(t) + cu(t) (51)
that are calibrated in way that+0
h2c(x)dx = mink,c
(52)
under the condition
wmax Wo (53)with Wo any given upper bound on the control force.
In Fig. 5 the controlled response with a linear control, whose coefficients
(k = 572.2 kg/cm, c = 32.3 kgs/cm, obtained for Wo = 1/4 of the forcing functionenergy) are calibrated according to the above norm procedure, added on the free
component of the oscillations (with p() = 675 kg/cm and q() = 0.375 kgs/cm),
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Independent Harmonic Control for Structural Engineering 311
-2E
-3
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(b)
Fig. 6. Response (a) and control force (b) only by norm control, with optimal coefficientsk = 572.2 kg/cm, c = 32.3 kgs/cm, obtained for Wo = 1/4 of the forcing function energy.
is plotted; it shows that, when necessary, it is possible to resort to such provision
in order to compensate for the initial conditions effect, as one can observe by com-
parison with Fig. 4, where the additional control is absent.
Finally, from Fig. 6, where the results derived by the adoption of the only norm
control algorithm are reported, the convenience of adopting such a combined ap-
proach, both in terms of effectiveness and economy of the employed control energy,
can be appreciated.
4. Application to an Earthquake-Type Forcing Function
It is expected anyway, and it is confirmed by the numerical investigation (not
extensively reported here for sake of brevity), that when the active force has a
long duration (compared to the structure own period To) and has a initial build-
up phase in which its intensity increases gradually, the effect produced by initialconditions is largely attenuated.
The numerical investigation has been carried on considering the time-history
recorded at Sturno in the Campania region in Southern Italy, during the 23 Novem-
ber large earthquake in 1980. It is a quake characterised by a long duration (more
than 70 s) and two subsequent strong motion phases, spaced out by a decay phase,
in which the shake attains a peak acceleration of 0.225 g.
Its time-history and power spectrum are plotted in Fig. 7. In Fig. 7(b), it is
also reported the smoothed power spectrum S(), responding to the expression
resulting from the superposition of the two KanaiTajimi curves
S() = [G(|a a) + G(|b b)] (54)
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Independent Harmonic Control for Structural Engineering 313
where is a normalisation factor, yielding the equality of the total energy between
the actual and the smooth spectrum [Baratta and Zuccaro, 1999], and the param-
eters a, a, b, b are given in Fig. 7(b).The control strategy can be designed in way that the control action is more
intense at the frequency that can transmit more energy to the structure. This
scope can be achieved giving the function Co() the following expression
Co() =
ComS()|Ho()| if ComS()|Ho()| ComaxComax otherwise .
(56)
In Figs. 8 and 9, the results of the simulation are quoted assuming Com =
Comax = 0.9.
5. Conclusions
In the paper a procedure, able to optimise the control action with respect to the
foreseen forcing function and constrained by an upper bound on the control action,
has been developed. The procedure is based on the harmonic decomposition of
the forcing function and on the ideal distribution of the control action over the
frequency range in an optimal fashion with respect to the expected power density
distribution of the active force. The set up of the rationale allows to distribute thecontrol power over the frequency range, by simply assigning the function Co(),
in way to save the control power where the control action is expected to be less
effective. Furthermore the significant influence of the initial conditions on what one
could deduce if referring to the frequency domain has been deeply analysed. A wide
numerical investigation (here only partially reported) has been finally carried out,
demonstrating the effectiveness of the procedure, that allows a consistent response
mitigation with a minimum control energy employ.
References
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Baratta, A., Cennamo, C. and Voiello, G. [1998] Design of linear active control algorithmsfor non-stationary unknown forcing function, J. Struct. Control2, pp. 731.
Baratta, A. and Corbi, O. [1999] Improved control of structures by time-delayed algo-rithms, Proc. of the 5th International Conference on the Application of ArtificialIntelligence to Civil and Structural Engineering, Oxford, pp. 159168.
Baratta, A. and Corbi, O. [1999] Algoritmo di controllo per forzanti armoniche neldominio delle frequenze, Proc. of the 12nd Italian Congress of ComputationalMechanics, GIMC, Napoli, pp. 1116.
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Baratta, A. and Corbi, O. [2000] On the optimality criterion in structural control,Earthq. Engg. Struct. Dyn. 29, pp. 141157.
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