engineering applications of artificial...
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Engineering Applications of Artificial Intelligence 55 (2016) 47–57
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Engineering Applications of Artificial Intelligence
http://d0952-19
n CorrE-m
journal homepage: www.elsevier.com/locate/engappai
New discrete-time robust H2/H1 algorithm for vibration controlof smart structures using linear matrix inequalities
Zhijun Li a, Hojjat Adeli b,n
a School of Civil and Architecture Engineering, Xi’an Technological University, Xi’an 710032, Chinab Department of Civil, Environmental, and Geodetic Engineering, The Ohio State University, 470 Hitchcock Hall, 2070 Neil Avenue, Columbus, OH 43220, USA
a r t i c l e i n f o
Article history:Received 14 December 2015Received in revised form2 April 2016Accepted 16 May 2016
Keywords:Robust controlDiscrete-time controlH2/H1 controlParametric uncertaintiesSmart structuresVibration control
x.doi.org/10.1016/j.engappai.2016.05.00876/& 2016 Elsevier Ltd. All rights reserved.
esponding author.ail address: [email protected] (H. Adeli).
a b s t r a c t
In real structural systems, such as a building structure or a mechanical system, due to inherent structuralmodeling approximations and errors, and changeable and unpredictable environmental loads, thestructural response unavoidably involves uncertainties. These uncertainties can reduce the performanceof a control algorithm significantly and possibly make it unstable. In this paper, based on the theories ofthe Bounded Real Lemma and the linear matrix inequalities (LMI), a novel discrete-time robust H2/H1control algorithm is presented which not only reduces the structural peak response caused by externaldynamic forces but also is robust and stable in the presence of parametric uncertainties which is alwaysthe case in real-life structures. To facilitate practical implementation, the uncertainties of structuralparameters are considered in the time domain as opposed to the frequency domain. Compared withtraditional H1 control methods, the new control algorithm proposes a convenient design procedure tofacilitate practical implementations of active control of complex and large structural systems through theuse of a quadratic performance index and the LMI-based solution method. The effectiveness of the newdiscrete-time robust H2/H1 adaptive control algorithm is demonstrated using a three-story frame withactive bracing systems (ABS) and a ten-story frame with an active tuned mass damper (ATMD).
& 2016 Elsevier Ltd. All rights reserved.
1. Introduction
Reducing the peak response quantities such as displacementsand accelerations of structures subjected to external dynamicloads is of primary concern in design of large structures. The mostrecent design strategies focus on methods of structural vibrationcontrol (Adeli and Saleh, 1999; Adeli and Jiang, 2009; Adeli andKim, 2009). These methods are divided into passive control suchas the Tuned Mass Damper (TMD) (Gutierrez-Soto and Adeli,2013a; Andersson et al., 2015), semi-active control (Fisco andAdeli, 2011a), active control (Kim and Adeli, 2005d; Gutierrez-Sotoand Adeli, 2013b), and hybrid control (Kim and Adeli, 2005b,2005c; Fisco and Adeli, 2011b) method. Compared to the passivecontrol system, an active control system has advantages ofadaptability and performance. Moreover, semi-active and hybridcontrol strategies which are more practical in terms of im-plementation are always based on active control algorithms (El-Khoury and Adeli, 2013).
Over the past few decades many active control algorithms havebeen developed such as the linear quadratic regulator (LQR)
(Stavroulakis et al., 2006), linear quadratic Gaussian (LQG) (Wu andYang, 2000), sliding mode control (SMC) (Alli and Yakut, 2005; Pai,2010; Wang and Adeli, 2012, 2015a, 2015b), H1 control (Yang et al.,1996), proportional–integral–derivative (PID) control (Kang et al.,2009), model predictive control (Wang et al., 2015), parallel control(Li et al., 2014), and optimal control algorithm (Adeli and Saleh, 1997;Saleh and Adeli, 1997, 1998a, 1998b, Li et al., 2015). Adeli and Saleh(1998, 1999) present an integrated control and optimization strategyfor design of both civil structures and control system. For solution ofthe integrated control and optimization Saleh and Adeli (1994) pre-sent parallel algorithms on high-performance parallel machines(Adeli and Kamal, 1993) and supercomputers (Adeli and Soegiarso,1999). A review of recent advances on vibration control of structuresunder dynamic loading is presented by Khoury and Adeli (2013)
In real structures, due to inherent structural modeling ap-proximations and errors, and changeable and unpredictable en-vironmental loads, the structural system response unavoidablyinvolves uncertainties. These uncertainties can reduce the perfor-mance of a control algorithm and possibly make it unstable. In thepresence of structural parameters uncertainties traditional controlmethods do not provide the stability and robustness needed foreffective reduction of the structural response under unknowableand varying external dynamic loading conditions. They canaffect the structure adversely when the frequency of external
Z. Li, H. Adeli / Engineering Applications of Artificial Intelligence 55 (2016) 47–5748
disturbance is close to the natural frequency of the structure. Kimand Adeli (2004a,b) developed a hybrid feedback-least meansquare (LMS) algorithm and an improved wavelet-hybrid feed-back-LMS algorithm to suppress both steady and transient vibra-tions effectively. They demonstrated their effectiveness for sig-nificant vibration reduction in both irregular highrise building(Kim and Adeli, 2005a) and bridge structures (Kim and Adeli,2005d).
Based on the merits of neural networks and fuzzy logic meth-ods for system identification (Boutalis et al., 2013; Rigatos, 2013),Jiang and Adeli (2008a, 2008b) present a dynamic fuzzy waveletneuroemulator to predict the structural response, and then obtainthe optimal control forces using the genetic algorithm. Amini andZabihi-Samani (2014) present a time varying wavelet-based poleassignment (WPA) method to control the seismic response of abuilding structure. However, the uncertainties of the structuralparameters, such as stiffness, damping, and mass, are not takeninto account in any of these methods.
A key issue in practical implementation of the vibration controltechnology is the robustness of the control algorithm in the presenceof uncertainty which is the subject of this paper. H1 control is apopular control algorithm in the area of robust control that requiresthe solution of the Riccati equation (Saleh and Adeli, 1997; Zhou andDoyle, 1998) which is time consuming in terms of computing. A fewresearchers have used the H1 control for vibration control of struc-tures based on the continuous time system. Yang et al. (1996) showthat the H1 control method is effective for reducing the seismic re-sponse of building structures but do not consider uncertainties ofstructural parameters in the formulation. Calise and Sweriduk (1998)describe an H1 control method based on the frequency domain butthe uncertainties of structural parameters cannot be described easilyin the frequency domain. Wang et al. (2004) present an H1 con-troller taking into account the norm-bounded uncertainties ofstructural parameters through solving a Riccati equation andchoosing a set of flexible scalars. Simulation results show the effec-tiveness of the algorithm with a perturbation of 10% for mass, stiff-ness and damping coefficients. However, solving the resulting Riccatiequation with excessive flexible scalars can be problematic and theperformance of the control algorithm with a relatively high pertur-bation can deteriorate. The aforementioned H1 control methods allrequire the solution of the Riccati equation.
An effective approach for solution of the Riccati equation is ap-plication of the linear matrix inequalities (LMI) (Gahinet and Apkar-ian, 1994). Du et al. (2004), Wu et al. (2006), and Du et al. (2011) usedan LMI-based solution approach for design of H1 controllers. Chang(2005) designed a mixed H2/H1 control algorithm based on the fre-quency domain and applied it to the gap control of the electric dis-charge machines. Guimaraes et al. (2007) proposed an immune-basedH2/H1 controller. Yang et al. (2014) used a mixed H2/H1 control al-gorithm to control the temperature of the four-zone split inverter airconditioners. These methods, however, have shortcomings becauseeither they do not consider the structural parameter uncertainties intheir formulation or deal with them in the frequency domain whichmakes their application to real structures difficult.
Since real-life control systems are modeled as a discrete-timesystem, an operational discrete-time robust control algorithm is ofparamount importance for practical implementation of vibrationcontrol of structures with uncertainties. In this paper, a noveldiscrete-time robust H2/H1 control algorithm is presented for vi-bration control of structures subjected to dynamic loading such asstrong ground motions taking into account the uncertainties inmodeling the structure based on the theories of the Bounded RealLemma (Zhou and Doyle, 1998) and using the LMI approach (Boydet al., 1994). To facilitate practical implementation, the un-certainties of structural parameters are considered in the timedomain as opposed to the frequency domain. The H1 approach is
employed to achieve stability. To increase the effectiveness of thecontrol algorithm it is integrated with an H2 algorithm where aquadratic performance index is used to evaluate and compare theperformance of the control system. The effectiveness of the newdiscrete-time robust H2/H1 control algorithm is demonstratedusing a three-story frame with active bracing systems (ABS) and aten-story frame with an active tuned mass damper (ATMD).
2. Modeling of uncertainty in the control equations
The dynamic equation of motion for an n-Degree-of-Freedom(DOF) structure subjected to one-dimensional ground acceleration
( )w t and active control forces is described as follows:
¨ + + = − ¯ ( ) + ( ) ( )w t tMX CX KX MI B U 1s
where X is the column vector of displacements relative to theground, M¼diag[m1, m2, …, mn] is the diagonal mass matrix, Kand C are n�n stiffness and damping matrices, respectively, Bs isthe n� r control device location matrix, r is the number of activecontrol devices, ( )tU is the r-dimensional vector of control forces,and I is an n�1 vector whose values are all equal to one.
In this research uncertainties of the structural parameters aredescribed through perturbations of the parameters. As such, thedynamic equation of the structure with uncertainties can be de-scribed in the following manner:
Δ Δ Δ
Δ Δ
( + ) ¨ + ( + ) + ( + )
= − ( + )¯ ( ) + ( + ) ( ) ( )w t t
M X C X K X
M I B U 2
M C K
M s BS
where ΔM, ΔK , ΔCand ΔBS are corresponding perturbations pre-scribed using a preselected scalar δ< <0 1i so that
Δ δ< ≤ <M0 / 1Mi i i which can guarantee Δ+M M is non-singular.Consequently, the uncertainty ΔM satisfies the following boundwhere δ is an n�n diagonal matrix with terms δi in the diagonal(δi is chosen according to amplitude of uncertainty ΔM):
Δ δ‖ ‖ ≤ ‖ ‖ < ( )−M 1 3M1
To avoid the problematic inversion of Δ+M M, consider twodifferent scalar matrices δ and δ′, and assume the following re-lationship holds: δ δ( + )( + ′) =I I I. Then, Eq. (2) can be rearrangedas follows:
δ Δ δ Δ
δ Δ
¨ + ( + ′) ( + ) + ( + ′) ( + )
= − ¯ ( ) + ( + ′) ( + ) ( ) ( )
− −
−w t t
X I M C X I M K X
I I M B U 4
1C
1K
1s BS
Following the state space representation, Eq. (4) can be ex-pressed in the following form:
Δ Δ ( ) = ( + ) ( ) + ( + ) ( ) + ( ) ( )t t t w tZ A A Z B B U H 5
where
δ δΔ =
− −− −
⎡⎣⎢
⎤⎦⎥A
M M0 0
K C1 1 ,
δΔ = −
⎡⎣⎢⎢
⎤⎦⎥⎥B
M0
Bs1 , ( ) =
⎡⎣⎢
⎤⎦⎥tZ X
X, =
− −− −
⎡⎣⎢
⎤⎦⎥A I
M K M C0
1 1,
= −
⎡⎣⎢
⎤⎦⎥B
M B0
s1 , { }= −
⎡⎣⎢⎢
⎤⎦⎥⎥H
01
,
δ δ Δ δ= ( + ′) + ′I KK K , δ δ Δ δ= ( + ′) + ′I CC C , δ δ Δ δ= ( + ′) + ′BIBS BS s
Because the control system expressed by Eq. (5) is linear andtime-invariant, the following analytical solution is obtained (Zhouand Doyle, 1998):
∫
∫
τ τ
τ τ
( ) = ( ) + ( + Δ ) ( )
+ ( )( )
τ
τ
( +Δ )( − ) ( +Δ )( − )
( +Δ )( − )
t e t e d
e w d
Z Z B B u
H6
t t
t
tt
t
tt
A A A A
A A
00
0
0
Z. Li, H. Adeli / Engineering Applications of Artificial Intelligence 55 (2016) 47–57 49
During real-time control, the control forces are calculated onceevery sampling period, T. The external excitations are digitizedusing the same sampling period, T, which means values betweentwo consecutive sampling instants are assumed to be constant.The control forces are also constant between two consecutivesampling instants. Letting =t kT0 and = ( + )t k T1 , Eq. (6) becomes
∫
∫
τ
τ
[( + ) ] = ( )
+ ( + Δ ) ( )
+ ( ) ( )
τ
τ
( +Δ )
( + )( +Δ ) ( + ) −
( + )( +Δ ) ( + ) −
⎡⎣ ⎤⎦
⎡⎣ ⎤⎦
k T e kT
e kT d
e w kT d
Z Z
B B U
H
1
7
T
kT
k Tk T
kT
k Tk T
A A
A A
A A
11
11
Assuming η τ= ( + ) −k T1 , Eq. (7) can be written as
∫
∫
η
η
[( + ) ] = ( ) + ( + Δ ) ( )
+ ( ) ( )
η
η
( +Δ ) ( +Δ )
( +Δ )
k T e kT e kT d
e w kT d
Z Z B B U
H
1
8
TT
T
A A A A
A A
0
0
Neglecting the higher order infinitesimals, the followingquantities are defined:
∫ ∫
∫
η η
η
= = =
Δ = Δ Δ = Δ( )
η η
η⎛⎝⎜
⎞⎠⎟
e e d e d
e T e d
A B B H H
A A B B
, , ,
,9
TT T
TT
dA
dA
dA
dA
dA
0 0
0
where the subscript d refers to the discrete time system. Using theaforementioned definitions Eq. (8) is transformed to
( + ) = ( + Δ ) ( ) + ( + Δ ) ( ) + ( ) ( )k k k w kZ A A Z B B U H1 10d d d d d
The controlled output is obtained from
( ) = ( ) + ( )( )
⎡⎣⎢
⎤⎦⎥
⎡⎣⎢
⎤⎦⎥k k kz
QZ R U
00 11
1 1/2
1/2
where ≥Q 0 and >R 0are weighting matrices. The measureddisplacement and velocity output Ys is obtained by
Γ( ) = ( ) = ( ) + ( ) ( )k k C k C kY Z X X 12s d v
where Cd is the location matrix of displacement sensors and Cv isthe location matrix of the velocity sensors. They can be placedcollectively in Γ = ⎡⎣ ⎤⎦C Cd v
3. New discrete-time robust H2/H1 algorithm for vibrationcontrol of structures using linear matrix inequalities
3.1. Definition of admissible uncertainties
Using Eqs. (3) and (5), δK , δC , and δBS can be written in thefollowing form:
δ = ( )L F E 13K k k k
δ = ( )L F E 14C c c c
δ = ( )L F E 15B b b bS
where ‖ ‖ ≤F 1k , ‖ ‖ ≤F 1c , ‖ ‖ ≤F 1b . Matrices Lk, Ek, Lc, Ec, Lb, Eb areregarded as constant and describe how the uncertain parametersin Fk, Fc, Fb enter the original matrices C, K and BS, respectively. Ifthe uncertainties in structural system represented by Eq. (10) sa-tisfy the conditions of Eqs. (13–15) they are said to be admissible.
Substituting Eqs. (13–15) into Eq. (5), ΔA and ΔB can be re-
written as
Δ = ( )Δ = ( ) ( )
⎫⎬⎭tt
A DF EB DF E 16
d
d 2
where ( )∫ η= − η− − −
⎡⎣⎢⎢
⎡⎣⎢
⎤⎦⎥
⎡⎣⎢
⎤⎦⎥⎤⎦⎥⎥Te e dD
M L M L M L0 0 0T
k c
T
b
A A1 1 0 1 , =
⎡
⎣⎢⎢⎢
⎤
⎦⎥⎥⎥
FF
FF
0 00 00 0
k
c
b
,
=⎡
⎣⎢⎢⎢
⎤
⎦⎥⎥⎥
EE
E0
00 0
k
c1 , = −⎡
⎣⎢⎢⎢
⎤
⎦⎥⎥⎥
EE
00
b
2
3.2. Definition of stability and performance index
For the system described by Eq. (10) containing all the ad-missible uncertainties, if there exists a control law, ( ) = ( )k kU K Zu ,which satisfies the following three conditions:
(i) The closed loop system represented by Eq. (10) is asymp-totically stable.
(ii) The measured signals ( )w k and ( )kYs are bounded byγ‖ ( )‖ <∞T sY ws , where disturbance attenuation parameter γ > 0 is a
given constant.(iii) The performance index J is represented by
{ }= ( ) ( ) ≤ ¯( )→∞
J k k Jz zsup limE17F t
1T
1
where sup indicates the upper bound, J is a known constant de-fining the upper boundary of H2 performance index, and { ⋅}Edenotes the expectation operator matrix, then ( ) = ( )k kU K Zu isconsidered to be a robust control algorithm with a guaranteed H2
performance index J .
3.3. New control law
Substituting the control law ( ) = ( )k kU K Zu into Eqs. (10–12) willtransform them to
( + ) = ¯ ( ) + ( ) ( )k k w kZ A Z H1 18d d1
( ) = ( ) ( )k C kz Z 19d1 1
where ¯ = +A A DFEd d1 1 , = +⎡⎣⎢
⎤⎦⎥
⎡⎣⎢
⎤⎦⎥C
QR K
0
0d u1 1/2
1/2, = +A A B Kd d d u1 ,
= +E E E Ku1 2 .Eq. (12), (18), and (19) constitute the new control algorithm
whose robustness will be proven mathematically in the nextsection. If the matrix Ad1 containing the control gains is stable thenthe performance index J can be expressed as
{ }= ˜( )
J H PHsupTrace20F
d dT
where ˜ = ˜ ≥P P 0T is obtained by solving the following discrete-time Lyapunov equation:
¯ ˜ ¯ − ˜ + = ( )A PA P C C 0 21d d d d1T
1 1T
1
3.4. Proof of stability of the new control algorithm
Before the proof, the following mathematical lemmas are listed.
Lemma 1. Consider a given matrix =*
⎡⎣⎢
⎤⎦⎥S
S SS
11 12
22with =S S11 11
T ,
=S S22 22T then the following conditions hold (Boyd et al., 1994).
(1) <S 0,
(2) <S 022 , − <−
S S S S 011 12 221
12T .
Lemma 2. Suppose YC and ZC are real rectangular matrices and XC is areal square matrix of arbitrary dimensions and =X XC C
T . Then, (Yu and
Z. Li, H. Adeli / Engineering Applications of Artificial Intelligence 55 (2016) 47–5750
Gao, 2001) + + <X Y FZ Z F Y 0C C C C CT T T for all F satisfying ≤F F IT , if and
only if there exists a scalar η40 such that η η+ + <−X Y Y Z Z 0C C C C CT 1 T .
Theorem 1. For a given scalar γ > 0, Ad1 is asymptotically stable andγ‖ ( )‖ <∞T sY ws for the uncertain structural system represented by Eq.
(10) with (16), if a positive definite matrix = >P P 0T guarantees that
( )γ α α Γ Γ¯ ¯ − + ¯ − ¯ +
+ < ( )
− − −A PA P A PH I H PH H PA
C C 0 22
d d d d d d d d
d d
1T
1 1T 2 1 T 1 T
11 T
1T
1
and γ α − >− I H PH 0d d2 1 T for an existing constantα > 0.
For the positive definite matrix = >P P 0T , it should also satisfythe following condition
≤ ˜ ≤ ( )P P0 23
where P can be obtained through solving the discrete-time Lya-punov Eq. (21).
Proof. According to the Bounded Real Lemma (Zhou and Doyle,1998), Ad1 is asymptotically stable and γ‖ ( )‖ <∞T sY ws for an un-certain structural system represented by Eq. (10), if and only if amatrix ¯ = ¯ >P P 0T exists such that
( )γ Γ Γ¯ ¯ ¯ − ¯ + ¯ ¯ − ¯ ¯ ¯ + < ( )−
A PA P A PH I H PH H PA 0 24d d d d d d d d1T
1 1T 2 T 1 T
1T
and γ − ¯ >I H PH 0d d2 T is satisfied.
Eq. (24) can be expressed in another form:
( )γ
α
Γ Γ¯ ¯ ¯ − ¯ + ¯ ¯ − ¯ ¯ ¯ +
+ < ( )
−A PA P A PH I H PH H PA
C C 0 25
d d d d d d d d
d d
1T
1 1T 2 T 1 T
1T
1T
1
where α > 0 is a given scalar.Define a new variable α= ¯−P P1 in Eq. (25). Pre- and post-
multiplication of Eq. (25) by constant α− I1 results inγ α − >− I H PH 0d d
2 1 T and Eq. (22). Define the ma-
trix ( )γ α Γ Γ= ¯ − ¯ +− −M A PH I H PH H PAd d d d d d d1T 2 T 1 T
11 T . If Eq. (22) and
γ α − >− I H PH 0d d2 1 T are satisfied, it is deduced that ≥M 0d . Sub-tracting Eq. (21) from Eq. (22), the following inequality is ob-tained: ( ) ( )¯ − ˜ ¯ − − ˜ + <A P P A P P M 0d d d1
T1 . From the conditions
that ≥M 0d and matrix Ad1 is stable it is concluded − ˜ ≥P P 0.
3.5. Computation of control forces via LMI
The computation of the robust H2/H1 control forces( ) = − ( )k kU K Zu in Theorem 1 is realized by using the LMI method
in the following Theorem 2.
Theorem 2. For a given scalarγ > 0, if and only if there exists scalarsα > 0 and β > 0, such that a positive definite matrix = >N N 0T and amatrix Y satisfy the following condition:
( )
αγ α
ββ
α
Γ− ( + ) ( + ) ( ) +
* −
* * − +* * * −* * * * −* * * * * −
<
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎛⎝⎜⎜
⎡⎣⎢⎢
⎤⎦⎥⎥
⎞⎠⎟⎟
⎛⎝⎜⎜
⎡⎣⎢
⎤⎦⎥
⎞⎠⎟⎟
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥ 26
N A N B Y E N E Y NQ
N R Y
I H
N DDI
II
00
0
0 0 0
0 0 00 0
0
0
d d
d
T1 2
T T1/2
T1/2
T
2 T
T
then the following results hold:
(1) the uncertain structural system (10) is robust with the upperboundary of H2 performance index, { }¯ = −J H N HTrace d d
T 1 ;
(2) a stable H2/H1 control force vector U with the guaranteed H2
performance index J is obtained by
( ) = ( ) = ( ) ( )−k k kU K Z YN Z 27u1
where the notation * is used to sign transpose of a block matrix.
Proof. Based on Lemma 1, Eq. (22) and γ α − >− I H PH 0d d2 1 T are
satisfied, if and only if there exists
α
α γ
Γ Γ− + + ¯
¯ −
<
( )
−
−
−
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
P C C A
I H
A H P
0
0 0
28
d d d
d
d d
1 T1
T1 1
T
1 2 T
11
Define a matrix by
α
α γ
Γ Γ=
− + +
− ( )
−
−
−
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥S
P C C A
I H
A H P
0
0
29
d d d
d
d d
1 T1
T1 1
T
1 2 T
11
Using the relation ¯ = +A A DFEd d1 1 and matrix S in Eq. (29), Eq.(28) is written in the following form:
+ + <( )
⎡
⎣⎢⎢
⎤
⎦⎥⎥ ⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦
⎡
⎣⎢⎢
⎤
⎦⎥⎥S
DF E E F
D
00 0 0 0 0
00 0
30
T T
T
According to Lemma 2, Eq. (30) is satisfied for all F satisfying≤F F IT , if and only if there exists a scalar β > 0 such that
β β+ + <( )
−⎡
⎣⎢⎢
⎤
⎦⎥⎥⎡⎣ ⎤⎦
⎡
⎣⎢⎢
⎤
⎦⎥⎥⎡⎣ ⎤⎦S
DD
EE
00 0 0 0
00 0 0
31
T 1
T
is satisfied.According to Lemma 1 and Eq. (29), Eq. (31) can be re-written
as
α γ
ββ
α
Γ−
* −* * − +* * * −* * * * −* * * * * −
<
( )
−
−
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥
P A E C
I H
P DDI
II
0
0 0 0
0 0 00 0
0
0
32
d d
d
1T T T
1T
1 2 T
1 T
Through pre- and post-multiplication of Eq. (32) by the
{ }α−P I I I I Idiag , , , , ,1 , and defining the new variable = −N P 1
and =Y K Nu , one can obtain = −K YNu1and the following in-
equality
αγ α
ββ
α
Γ−
* −* * − +* * * −* * * * −* * * * * −
<
( )
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥
N NA NE N NC
I H
N DDI
II
0
0 0 0
0 0 00 0
0
0
33
d d
d
1T T T
1T
2 T
T
As matrices Ad1, E, Γ, Cd1 are expressed by another equal formaccording to the new control algorithm defined by Eq. (12), (18),and (19), Eq. (25) is obtained from Eq. (33).
Fig. 3. Acceleration responses of the third story of Example 1.
Z. Li, H. Adeli / Engineering Applications of Artificial Intelligence 55 (2016) 47–57 51
4. Applications
In order to verify the effectiveness and robustness of the newrobust H2/H1 control method, two examples are presented.
4.1. Example 1
This is a three Degree-of-Freedom (DOF) shear frame structurewith an active bracing system (ABS) on each story shown in Fig. 1taken from the literature (Ou, 2003). The values for the structuralparameters are = ×m 4 10i
2metric ton, = ×k 2 10i8 N/m, and
ξ = 5% ( = − )i 1 3 . The sampling period is T¼0.02 s. The N–Scomponent of the 1940 EI Centro earthquake record in the Im-perial Valley scaled to a maximum ground acceleration of 0.2 gover a duration of 8 s is used as the input excitation.
First, the H1 control method of Yu and Gao (2001) is appliedfor the sake of comparison. Using γ = 1.0 and Γ = I6 (3 displace-ment and 3 velocity sensors), the control gain matrix Ku is ob-tained. The displacement and acceleration responses of the thirdstory relative to the ground are shown in Figs. 2 and 3, respec-tively, for both controlled and uncontrolled structure. The re-sponse quantities for the nominal (original) structural system(with ABS control devices) are designated as “H-1(0%)”. It is ob-served that the traditional H1 control algorithm is relatively ef-fective for the nominal structural system.
Next, the values of the stiffness of all the stories of the structureand damping coefficient are reduced by 20%. The displacementand acceleration responses of the third story using the traditionalH1 control algorithm are presented in Figs. 4 and 5, respectively. Itis observed that the control algorithm becomes unstable when thestructure is subjected to a large amplitude parametric uncertainty.It is concluded that ignoring the parametric uncertainties in the
ABS
ABS
ABS
m2
m1
m3
Fig. 1. Example 1.
Fig. 2. Displacement responses of the third story of Example 1.
design of control algorithm may result in the unstable controlsystem only using the unadjusted control gain.
Next, the new robust H2/H1 controller presented in this paperis applied. The following uncertainty values are used for stiffness,damping, and location of control forces/actuators, respectively:
Δ = K0.2K , Δ = C0.2C , Δ = B0.10B SS .LQR is a special case of the H2 method. For a relatively accurate
comparison of the robust stability of the traditional H1 controllerand the new H2/H1 controller, the weighing matrices Q and R, arechosen based on the same method used in the LQR method (Stav-roulakis et al., 2006) and the following weighing matrices are used:
{ }= × × × × × ×Q diag 8 10 8 10 2 10 4 10 4 10 4 108 8 8 5 5 5 , =R
{ }× × ×− − −diag 3 10 3 10 3 104 4 4 .Using γ = 3.16, α = 0.0085, β = × −4.4079 10 13, = =E E Ik c 3,
Γ = I6, and { }=E diag 1 1 1b , the control gain matrix Ku is ob-tained using Eqs. (25) and (26):
Fig. 4. Displacement response of the third story of Example 1 with a 20% reductionin the stiffness and damping coefficients using H1 algorithm.
Fig. 5. Acceleration response of the third story of Example 1 with a 20% reductionin the stiffness and damping coefficients using H1 algorithm.
Z. Li, H. Adeli / Engineering Applications of Artificial Intelligence 55 (2016) 47–5752
Displacement and acceleration responses of the third storyusing the new robust control algorithm are shown in Fig. 6 forthree cases of uncontrolled, the nominal structural system
Fig. 6. Displacement (a) and acceleration (b) responses of the third story of ex-ample 1 for three cases of uncontrolled, the nominal structural system designatedas RH1(0%), and the case of reduced stiffness and damping coefficients by 20%designated as RH1(�20%) using the new robust control algorithm.
Fig. 7. The maximum inter-story drifts (a), maximum floor accelerations (b), maximumstructure for two sets of weighing matrices [RH1(0%) and RH2(0%)], and the structures wmatrices [RH1(�20%) and RH2(�20%)], and the performance index versus the sum of
designated as RH1(0%), and the case of reduced stiffness anddamping coefficients by 20% designated as RH1(�20%). The resultsshow that the new robust H2/H1 control algorithm is robust underlarge parameter uncertainties.
The effectiveness of the new control algorithm can be improvedby changing the weighing matrices. To show this point, a secondset of values are used for the weighing matrices as follows:
{ }= × × × × × ×Q diag 8 10 8 10 2 10 4 10 4 10 4 108 8 8 5 5 5 ,
{ }= − − −R diag 10 10 104 4 4 . Using γ = 3.16, α = 0.0061,β = × −1.1834 10 13, = =E E Ik c 3, Γ = I6, and { }=E diag 0.5 0.5 0.5b ,Eqs. (25) and (26) result in the another control gain matrix Ku.
The maximum inter-story drifts, maximum floor accelerations,and maximum actuator/control forces for the five cases of un-controlled structure, nominal structure for two sets of weighingmatrices [RH1(0%) and RH2(0%)], and the structures with 20% re-duction in the stiffness and damping coefficients for two sets ofweighing matrices [RH1(�20%) and RH2(�20%)] are displayed inFig. 7(a), (b), and (c), respectively. It is observed that using rea-sonable values of controller parameters the controller can result in asubstantial reduction in the response of the structure. Fig. 7(d) shows the value of the performance index employed in theproposed controller versus the total sum of the control forces. Itincreases monotonically with the total required control forces. It is ameasure of the total energy imparted onto the building structure.
4.2. Example 2
This example, is a 10-story 10-DOF shear frame structure withan active tuned mass damper (ATMD) control system on the topfloor taken from the literature (Amini et al., 2013; Amini and
actuator/control forces (c) for the five cases of uncontrolled structure, nominalith 20% reduction in the stiffness and damping coefficients for two sets of weighingcontrol forces.
10
9
2
1
kTMDcTMD
mTMD
Actuator
Fig. 8. Example 2.
Z. Li, H. Adeli / Engineering Applications of Artificial Intelligence 55 (2016) 47–57 53
Zabihi-Samani, 2014) (Fig. 8). It is used to demonstrate the effec-tiveness of the new control algorithm in a hybrid control system.The mass of each story is 10 metric ton. The stiffness of each storyis ×2 103 kN/m. The modal damping ratio of each mode is as-sumed to be 2%. The mass of the secondary ATMD system is as-sumed to be 3% of the total mass of the building structure. Itsdamping ratio is 7%. Its modal circular frequency, 2.11 rad/s, ischosen to be nearly identical to the fundamental frequency of themain structure. The structure is subjected to the 1995 Kobeearthquake record in Hanshin, Japan scaled to a maximum groundacceleration of 0.2 g over a duration of 20 s.
Two different levels of parametric uncertainties are considered.First, a perturbation of 10% is used for stiffness and dampingcoefficients, and location of control forces/actuators as follows:
Δ = K0.1K , Δ = C0.1C , Δ = B0.10B SS .The following weighing matrices are used:Q ¼ diag{1000, 1000, 1000, 1000, 1000, 1000, 1000, 1000, 1000,
Table 1Maximum response quantities for the uncontrolled case, TMD, and RH1for Example 2.
Floor Uncontrolled (cm) TMD (cm) RH1(cm)
Nominal
1 3.62 3.49 3.202 3.21 3.03 2.733 2.99 2.79 2.394 2.54 2.39 1.975 2.09 2.01 1.626 2.41 2.14 1.737 2.40 2.27 1.958 2.72 2.66 1.909 2.56 2.47 1.5510 1.58 1.50 0.86Control force (kN) 28.58Performance index J (103) 4.09
1000, 0.001, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0.001}, = × −R 0.5 10 5.Using γ = 1.0, α = × −1.7266 10 8, β = × −1.9303 10 10,= =E E Ik c 11, Γ = I22, and Eb¼[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 1], one can obtain the corresponding control gainmatrix Ku via Eqs. (25) and (26).
Next, a perturbation of 20% is used for stiffness and dampingcoefficients and 10% for the location of control forces/actuators asfollows:
Δ = K0.2K , Δ = C0.2C , Δ = B0.10B SS .The same weighing matrices are used as in the previous case.
Using γ = 1.0, α = × −2.7891 10 8, β = × −2.5994 10 11, = =E E Ik c 11,Γ = I22, and Eb¼[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,0, 1], the control gain matrix Ku different from the aforementionedis obtained via Eqs. (25) and (26).
The maximum inter-story drifts and maximum actuator/controlforces for the three cases of uncontrolled structure, controlled withthe passive TMD, and five cases of ATMD for two aforementionedperturbation levels (RH1 and RH2) are presented in Tables 1 and 2,respectively. To show the robustness and stability of the new H2/H1control algorithm, the RH1 case is divided into five sub-cases of thenominal structural system, the case of reduced stiffness, dampingcoefficients and control forces by 10% designated as “�10%K, C and�10%BS”, the case of increased stiffness and damping coefficientsby 10% and reduced control forces by 10% designated as “10%K, Cand �10%BS”, the case of reduced stiffness and damping coefficientsby 10% designated as “�10%K, C,” and the case of increased stiffnessand damping coefficients by 10% designated as “10%K, C”. FromTables 1 and 2, it is observed that a) the ATMD control system ismore effective than the TMD control system for suppressing thepeak response, b) the control system is stable even for relativelylarge perturbations of the parameters in the order of 20%, c) theuncertainties in control forces have a small effect on the controlledresponse, and d) there exists a proportional relationship betweenthe H2 performance index J and the maximum control force.
Displacement and acceleration responses of the 10th floor for fourcases of uncontrolled, TMD control, and nominal structure for twosets of control gain matrices (RH1 and RH2) using the new robustcontrol algorithm are displayed in Figs. 9 and 10, respectively. Thesefigures show clearly that the hybrid ATMD control based the newrobust H2/H1 control algorithm is more effective in reducing thepeak responses compared with the passive TMD control.
Control force response histories using the new robust controlalgorithm are shown in Fig. 11(a) and (b) for three cases of thenominal structure designated as RH1(0%), the case of increasedstiffness and damping coefficients by 10% designated as RH1(10%),and the case of reduced stiffness and damping coefficients by 10%designated as RH1(�10%). It is shown clearly in Fig. 11(a) and (b)that there is a better robust stability for the ATMD control system
�10%K, C and �10%BS 10%K, C and �10% BS �10% K, C 10% K, C
3.20 2.97 3.20 2.972.50 2.73 2.50 2.732.32 2.30 2.32 2.301.96 1.96 1.96 1.961.59 1.77 1.59 1.771.86 1.71 1.87 1.711.98 2.10 1.99 2.121.90 1.96 1.90 1.971.54 1.56 1.54 1.570.90 0.83 0.90 0.83
22.30 21.64 30.04 27.872.49 2.39 4.52 3.89
Table 2Maximum response quantities for the uncontrolled case, TMD, and RH2 for Example 2
Floor Uncontrolled (cm) TMD (cm) RH2 (cm)
Normal �20% K, C and �10% BS 20% K, C and �10% BS �20% K, C 20% K, C
1 3.62 3.49 3.19 2.87 2.85 2.87 2.852 3.21 3.03 2.73 2.20 2.55 2.20 2.553 2.99 2.79 2.38 1.96 2.07 1.96 2.074 2.54 2.39 1.95 1.84 1.86 1.84 1.865 2.09 2.01 1.57 1.77 1.79 1.77 1.796 2.41 2.14 1.66 1.92 2.02 1.92 2.027 2.40 2.27 1.86 1.94 2.14 1.94 2.148 2.72 2.66 1.82 1.81 1.85 1.81 1.859 2.56 2.47 1.47 1.42 1.51 1.42 1.5110 1.58 1.50 0.85 0.98 0.82 0.98 0.82Control force (kN) 38.79 34.16 34.68 38.73 37.04Performance index J (103) 7.53 5.84 6.05 7.51 6.90
Fig. 9. Displacement responses of the 10th floor of example 2 for four cases ofuncontrolled, TMD control, and nominal structure for two sets of control gainmatrices (RH1 and RH2) using the new robust control algorithm: (a) uncontrolledand TMD; (b) uncontrolled and RH1; (c) uncontrolled and RH2.
Fig. 10. Acceleration responses of the 10th floor of example 2 for four cases ofuncontrolled, TMD control, and nominal structure for two sets of control gainmatrices (RH1 and RH2) using the new robust control algorithm: (a) uncontrolledand TMD; (b) uncontrolled and RH1; (c) uncontrolled and RH2.
Z. Li, H. Adeli / Engineering Applications of Artificial Intelligence 55 (2016) 47–5754
with the parametric uncertainties.The maximum response quantities concluding inter-storey dis-
placement, control force and H2 performance index J are shown inTable 2 for three cases of uncontrolled in column 2, the case of onlyinstalling a passive TMD control system on the top story of thebuilding structure designed as TMD in column 3, and the case of
installing an hybrid ATMD control system (active control force iscomputed by new robust control algorithm) on the top story of thebuilding structure designated as RH2 in columns 4–8. To show therobustness and stability of the new robust H2/H1 control algorithm,
Fig. 11. The control force histories for the six cases of nominal structure for twosets of control gain matrix [RH1(0%) and RH2(0%)], the structure with 20% reduc-tion in the stiffness and damping coefficients for two sets of control gain matrix[RH1(�20%) and RH2(�20%)], and the structure with 20% increase in the stiffnessand damping coefficients for two sets of control gain matrix [RH1(20%) and RH2(20%)]: (a) RH1(0%) and RH1(10%); (b) RH1(0%) and RH1(�10%); (c) RH2(0%) andRH2(20%); (d) RH2(0%) and RH2(�20%).
Z. Li, H. Adeli / Engineering Applications of Artificial Intelligence 55 (2016) 47–57 55
the RH2 case is divided as four sub cases of the nominal structuralsystem designated as “normal” in column 4, the case of reducedstiffness and damping coefficients by 20% and control forces by 10%designated as “�20%K, C and �10%BS” in column 5, the case of in-creased stiffness and damping coefficients by 20% and reduced con-trol forces by 10% designated as “20%K,C and �10%BS” in column 6,the case of reduced stiffness, damping coefficients by 20% designatedas “�20%K, C” in column 7, and the case of increased stiffness,damping coefficients by 20% designated as “20%K, C” in column 8.From results of the Table 2, it is also shown clearly that the ATMDcontrol system is more effective to suppress the peak response thanthe TMD control system. From simulation results shown in column 4–8 of Table 2, it shows that the control system can already keep betterrobust stability although there is a bigger value of uncertainties forthe structural parameters and compared with the results in column4–8 of Table 1, robustness of the active control algorithm decreases
slightly for a large amplitude parametric uncertainty. It is also shownthat the robust stability is influenced slightly by the errors betweenthe real control forces and computed ones.
Displacement and acceleration responses of the tenth storyusing the new robust control algorithm are shown in Figs. 9(c) and10(c) for two cases of uncontrolled, and the case of the originalstructure with ATMD control system in which the active controlforces are computed by the new robust control algorithm desig-nated as RH2. From Figs. 9(c) and 10(c), it is observed clearly thatthe new robust H2/H1 control method is very remarkable tosuppress the peak response although it contains large amplitudeparametric uncertainties.
The control force histories for the six cases of nominal structurefor two sets of control gain matrix [RH1(0%) and RH2(0%)], thestructure with 20% reduction in the stiffness and damping coeffi-cients for two sets of control gain matrix [RH1(�20%) and RH2(�20%)], and the structure with 20% increase in the stiffness anddamping coefficients for two sets of control gain matrix [RH1(20%)and RH2(20%)] are displayed in Fig. 11(a) to (d). In all the figuresthe differences are small and unnoticeable indicating the robust-ness and stability of the new control algorithm for the ATMDsystem with relatively large amplitude parametric uncertainties.
5. Conclusions
In vibration control of real structures, control forces are estimatedaccording to the finite element simulation model after the structuralstate responses are measured via signal sensors. Compared with realstructural parameters, The values of structural parameters such asstiffness and damping coefficients obtained from the simulationmodel may be higher or lower than those for the actual structures dueto modeling approximations and errors introduced by the finite ele-ment analysis, errors introduced during the vibration data collections,and uncertainties in determination of real structural parameters. Inview of the numerous inevitable uncertainties in structural parametersdetermination a new robust discrete time H2/H1 control algorithm ispresented in this paper for effective reduction of the response ofstructures under dynamic loadings such as seismic loadings. The dif-ference between the existing H2/H1 hybrid schemes and the newmethod is that either they do not consider the structural parameteruncertainties in their formulation or deal with them in the frequencydomain which makes their application to real structures difficult. Therobustness of the method was proven using theories of the BoundedReal Lemma and LMI and demonstrated using two example structuresfrom the literature. The new control algorithm is promising for prac-tical active control, semi-active, and hybrid control of large real-lifestructures. The authors are currently applying the new algorithm forhybrid control of a 24-story irregular reinforced concrete frame-shearwall built in Xian, China, in 2013.
A related frontier of structural engineering research is healthmonitoring of structures where significant advances have beenmade in recent years (Su et al., 2014; O’Byrne et al., 2014; Parket al., 2015). For example, Cha and Buyukozturk (2015) presentstructural damage detection using modal strain energy and amulti-objective genetic algorithm for optimization (Jia et al., 2014;Luna et al., 2014; Paris et al., 2015). Bursi et al. (2014) presenthealth monitoring of cable-stayed bridges. Yeum and Dyke (2015)present vision-based automated crack detection for bridge in-spection. Zhang et al. (2014) present fuzzy analytic hierarchyprocess synthetic evaluation models for structural health mon-itoring using the fuzzy logic concept (Forero Mendoza et al., 2014;Hsu, 2015). Čygas et al. (2015) present health monitoring of pa-vement structures. Torbol (2014) presents real time frequencydomain decomposition for structural health monitoring usinggeneral purpose graphic processing units. Story and Fry (2014)
Z. Li, H. Adeli / Engineering Applications of Artificial Intelligence 55 (2016) 47–5756
discuss structural impairment detection system using competitivearrays of artificial neural networks (Chira et al., 2014; Chakrabortyet al., 2014).
Another extension of the research presented in this paperwould be integration with health monitoring of structures. Such anintegration where health of the structure is monitored based onmeasurements from sensors and active or semi-active controllersapply appropriate forces to minimize the structural response usingthe control algorithm presented in this paper would be a leap inrealization of the full potential of large adaptive/smart structures.
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China under Grant number 51178388, State Scho-larship Found of China Scholarship Council under Grant number201208615016, Scientific Research and Development Found ofShaanxi Province under Grant number 2013K07-07, Special Sci-ence Foundation of Shaanxi Provincial Department of Educationunder Grant number 2013JK0612, President Foundation of Xi’anTechnological University under Grant number XAGDXJJ0919, andYouth Elite Project of Xi’an Technological University.
References
Adeli, H., Jiang, X., 2009. Intelligent Infrastructure: Neural Networks. CRC Press,United States.
Adeli, H., Kamal, O., 1993. Parallel Processing in Structural Engineering. ElsevierApplied Science, London.
Adeli, H., Kim, H., 2004a. Wavelet-hybrid feedback-least mean square algorithm forrobust control of structures. J. Struct. Eng. 130 (1), 128–137.
Adeli, H., Kim, H., 2009. Wavelet-Based Vibration Control of Smart Buildings andBridges. CRC Press, United States.
Adeli, H., Saleh, A., 1997. Optimal control of adaptive/smart bridge structures. J.Struct. Eng. 123 (2), 218–226.
Adeli, H., Saleh, A., 1998. Integrated structural/control optimization of large adap-tive/smart structures. Int. J. Solids Struct. 35 (28), 3815–3830.
Adeli, H., Saleh, A., 1999. Control, Optimization, and Smart Structures: High-Per-formance Bridges and Buildings of the Future. John Wiley & Sons, United States.
Adeli, H., Soegiarso, R., 1999. High-Performance Computing in Structural En-gineering. CRC Press, Boca Raton, Florida.
Alli, H., Yakut, O., 2005. Fuzzy sliding-mode control of structures. Eng. Struct. 27 (2),277–284.
Amini, F., Khanmohamadi Hazaveh, N., Abdolahi Rad, A., 2013. Wavelet pso-basedlqr algorithm for optimal structural control using active tuned mass dampers.Comput.-Aided Civ. Infrastruct. Eng. 28 (7), 542–555.
Amini, F., Zabihi-Samani, M., 2014. A Wavelet-based adaptive pole assignmentmethod for structural control. Comput.-Aided Civ. Infrastruct. Eng. 29 (6),464–477.
Andersson, A., O’Connor, A.J., Karoumi, R., 2015. Passive and adaptive dampingsystems for vibration mitigation and increased fatigue service life of a tied archrailway bridge. Comput.-Aided Civ. Infrastruct. Eng. 30 (9), 748–757.
Boutalis, Y., Christodoulou, M., Theodoridis, D., 2013. Indirect adaptive control ofnonlinear systems based on bilinear neuro-fuzzy approximation. Int. J. NeuralSyst. 23 (05), 1350022.
Boyd, S.P., El Ghaoui, L., Feron, E., Balakrishnan, V., 1994. Linear Matrix Inequalitiesin System and Control Theory. SIAM, India.
Bursi, O.S., Ceravolo, R., Kumar, A., Abbiati, G., 2014. Identification, model updatingand validation of a steel twin deck curved cable-stayed footbridge. Comput.-Aided Civ. Infrastruct. Eng. 29 (9), 703–722.
Calise, A.J., Sweriduk, G.D., 1998. Active attenuation of building structural responseusing robust control. J. Eng. Mech. 124 (5), 520–528.
Cha, Y.J., Buyukozturk, O., 2015. Structural damage detection using modal strainenergy and hybrid multi-objective optimization. Comput.-Aided Civ. Infra-struct. Eng. 30 (5), 347–358.
Chakraborty, R., Lin, C.T., Pal, N.R., 2014. Sensor (group feature) selection withcontrolled redundancy in a connectionist framework. Int. J. Neural Syst. 24 (6),1450021 (21 pages).
Chang, Y.-F., 2005. Mixed H2/H1 optimization approach to gap control on EDM.Control. Eng. Pract. 13, 95–104.
Chira, C., Sedano, J., Camara, M., Prieto, C., Villar, J.R., Corchado, E., 2014. A clustermerging method for time series microarray with product values. Int. J. NeuralSyst. 24 (6), 1450018 (18 pages).
Čygas, D., Laurinavičius, A., Vaitkus, A., Paliukaitė, M., Motiejūnas, A., Žiliūtė, L.,2015. Monitoring the mechanical and structural behavior of the pavement
structure using electronic sensors. Comput.-Aided Civ. Infrastruct. Eng. 30 (4),317–328.
Du, H., Lam, J., Sze, K.Y., 2004. Non-fragile H1 vibration control for uncertainstructural systems. J. Sound. Vib. 273 (4), 1031–1045.
Du, H., Zhang, N., Naghdy, F., 2011. Actuator saturation control of uncertain struc-tures with input time delay. Journal. Sound. Vib. 330 (18), 4399–4412.
El-Khoury, O., Adeli, H., 2013. Recent advances on vibration control of structuresunder dynamic loading. Arch. Comput. Methods Eng. 20 (4), 353–360.
Fisco, N., Adeli, H., 2011a. Smart structures: Part I—active and semi-active control.Sci. Iran. 18 (3), 275–284.
Fisco, N.R., Adeli, H., 2011b. Smart structures: Part II – hybrid control systems andcontrol strategies. Sci. Iran. – Trans. A: Civil. Eng. 18 (3), 285–295.
Forero Mendoza, L., Vellasco, M., Figueiredo, K., 2014. Intelligent multiagent co-ordination based on reinforcement hierarchical neuro-fuzzy models. Int. J.Neural Syst. 24 (8), 1450031 (20 pages).
Gahinet, P., Apkarian, P., 1994. A linear matrix inequality approach to H1 control.Int. J. Robust. Nonlinear Control. 4 (4), 421–448.
Guimaraes, F.G., Palhares, R.M., Campelo, F., Igarashi, H., 2007. Design of mixed H2
/H1 control systems using algorithms inspired by the immune system. Inf. Sci.177, 4368–4386.
Gutierrez Soto, M., Adeli, H., 2013a. Tuned mass dampers. Arch. Comput. MethodsEng. 20 (4), 419–431.
Gutierrez Soto, M., Adeli, 2013b. Placement of control devices for passive, semi-active, and active, vibration control of structures. Sci. Iran. – Trans. A: Civ. Eng.20 (6), 1567–1578.
Hsu, W.Y., 2015. Assembling a multi-feature EEG classifier for left-right motor datausing wavelet-based fuzzy approximate entropy for improved accuracy. Int. J.Neural Syst. 25, 8 (13 pages).
Jia, L., Wang, Y., Fan, L., 2014. Multiobjective bilevel optimization for production-distribution planning problems using hybrid genetic algorithm. Integr. Com-put.-Aided Eng. 21 (1), 77–90.
Jiang, X., Adeli, H., 2008a. Dynamic fuzzy wavelet neuroemulator for non-linearcontrol of irregular building structures. Int. J. Numer. Methods Eng. 74 (7),1045–1066.
Jiang, X., Adeli, H., 2008b. Neuro-genetic algorithm for non-linear active control ofstructures. Int. J. Numer. Methods Eng. 75 (7), 770–786.
Kang, M., Schonfeld, P., Yang, N., 2009. Computer-aided civil and infrastructureengineering. J. Adv. Transp. 24 (2), 109–119.
Kim, H., Adeli, H., 2004b. Hybrid feedback-least mean square algorithm for struc-tural control. J. Struct. Eng. 130 (1), 120–127.
Kim, H., Adeli, H., 2005a. Hybrid control of irregular steel highrise building struc-tures under seismic excitations. Int. J. Numer. Methods Eng. 63 (12), 1757–1774.
Kim, H., Adeli, H., 2005b. Hybrid control of smart structures using a novel wavelet-based algorithm. Comput.-Aided Civ. Infrastruct. Eng. 20 (1), 7–22.
Kim, H., Adeli, H., 2005c. Wind-induced motion control of 76-story benchmarkbuilding using the hybrid damper-tlcd system. J. Struct. Eng. 131 (12),1794–1802.
Kim, H., Adeli, H., 2005d. Wavelet hybrid feedback-LMS algorithm for robust controlof cable-stayed bridges. J. Bridge Eng. 10 (2), 116–123.
Li, J., Ji, H., Cao, L., Zang, D., Gu, R., Xia, B., 2014. Evaluation and application of ahybrid brain computer interface for real wheelchair control with multi-degreesof freedom. Int. J. Neural Syst. 24 (4), 1450014 (15 pages).
Li, Y., Liu, C., Gao, J.X., Shen, W., 2015. An integrated feature-based dynamic controlsystem for on-line machining, inspection and monitoring. Integr. Comput.-Ai-ded Eng. 22 (2), 187–200.
Luna, J.M., Romero, J.R., Romero, C., Ventura, S., 2014. Reducing gaps in quantitativeassociation rules: a genetic programming free-parameter algorithm. Integr.Comput.-Aided Eng. 21 (4), 321–337.
O’Byrne, M., Schoefs, F., Pakrashi, V., Ghosh, B., 2014. Regionally enhanced multi-phase segmentation technique for damaged surfaces. Comput.-Aided Civ. In-frastruct. Eng. 29 (9), 644–658.
Ou, J., 2003. Structural Vibration Control-Active, Semi-Active and Smart Control.Science, China.
Pai, M.-C., 2010. Sliding mode control of vibration in uncertain time-delay systems.J. Vib. Control.
Paris, P.C.D., Pedrino, E.C., Nicoletti, M.C., 2015. Automatic learning of image filtersusing cartesian genetic programming. Integr. Comput.-Aided Eng. 22 (2),135–151.
Park, K., Oh, B.K., Park, H.S., 2015. GA-based multi-objective optimization algorithmfor retrofit design on a multi-Core PC cluster. Comput.-Aided Civ. Infrastruct.Eng. 30, 12.
Rigatos, G.G., 2013. Adaptive fuzzy control for differentially flat mimo nonlineardynamical systems. Integr. Comput.-Aided Eng. 20 (2), 111–126.
Saleh, A., Adeli, H., 1994. Parallel algorithms for integrated structural/control op-timization. J. Aerosp. Eng. 7 (3), 297–314.
Saleh, A., Adeli, H., 1997. Robust parallel algorithms for solution of riccati equation.J. Aerosp. Eng. 10 (3), 126–133.
Saleh, A., Adeli, H., 1998a. Optimal control of adaptive building structures underblast loading. Mechatronics 8 (8), 821–844.
Saleh, A., Adeli, H., 1998b. Optimal control of adaptive/smart multistory buildingstructures. Comput.-Aided Civ. Infrastruct. Eng. 13 (6), 389–403.
Stavroulakis, G., Marinova, D., Hadjigeorgiou, E., Foutsitzi, G., Baniotopoulos, C.,2006. Robust active control against wind-induced structural vibrations. J. Wind.Eng. Ind. Aerodyn. 94 (11), 895–907.
Story, B.A., Fry, G.T., 2014. A structural impairment detection system using com-petitive arrays of artificial neural networks. Comput.-Aided Civ. Infrastruct. Eng.
Z. Li, H. Adeli / Engineering Applications of Artificial Intelligence 55 (2016) 47–57 57
29 (3), 180–190.Su, W.C., Huang, C.S., Chen, C.H., Liu, C.Y., Huang, H.C., Le, Q.T., 2014. Identifying the
modal parameters of a structure from ambient vibration data via the stationarywavelet packet. Comput.-Aided Civ. Infrastruct. Eng. 29 (10), 738–757.
Torbol, M., 2014. Real time frequency domain decomposition for structural healthmonitoring using general purpose graphic processing unit. Comput.-Aided Civ.Infrastruct. Eng. 29 (9), 689–702.
Wang, N., Adeli, H., 2012. Algorithms for chattering reduction in system control. J.Frankl. Inst. 349 (8), 2687–2703.
Wang, N., Adeli, H., 2015a. Self-constructing wavelet neural network algorithm fornonlinear control of smart structures. Eng. Appl. Artif. Intell. 41, 249–258.
Wang, N., Adeli, H., 2015b. Robust vibration control of wind-excited highrisebuilding structures. J. Civ. Eng. Manag. 21 (8), 967–976.
Wang, J., Li, S.E., Zheng, Y., Lu, X.Y., 2015. Longitudinal collision mitigation via co-ordinated braking of multiple vehicles using model predictive control. Integr.Comput.-Aided Eng. 22 (2), 171–185.
Wang, S.-G., Roschke, P.N., Yeh, H., 2004. Robust control for structural systems withunstructured uncertainties. J. Eng. Mech. 130 (3), 337–346.
Wu, J.-C., Chih, H.-H., Chen, C.-H., 2006. A robust control method for seismic
protection of civil frame building. J. Sound. Vib. 294 (1), 314–328.Wu, J.C., Yang, J.N., 2000. Lqg control of lateral–torsional motion of nanjing Tv
transmission tower. Earthq. Eng. Struct. Dyn. 29 (8), 1111–1130.Yang, J., Wu, J., Reinhorn, A., Riley, M., Schmitendorf, W., Jabbari, F., 1996. Experi-
mental verifications of H1 and sliding mode control for seismically excitedbuildings. J. Struct. Eng. 122 (1), 69–75.
Yang, Y.-B., Wu, M.-D., Chang, Y.-C., 2014. Temperature control of the four-zone splitinverter air conditioners using lmi expression based on Lqr for mixed H2/H1.Appl. Energy 113, 912–923.
Yeum, C.M., Dyke, S.J., 2015. Vision based automated crack detection for bridgeinspection. Comput.-Aided Civ. Infrastruct. Eng. 30 (10), 759–770.
Yu, L., Gao, F., 2001. Optimal guaranteed cost control of discrete-time uncertainsystems with both state and input delays. J. Frankl. Inst. 338 (1), 101–110.
Zhang, W., Sun, K., Lei, C., Zhang, Y., Li, H., Spencer Jr., B.F., 2014. Fuzzy analytichierarchy process synthetic evaluation models for the health monitoring ofshield tunnels. Comput.-Aided Civ. Infrastruct. Eng. 29 (9), 676–688.
Zhou, K., Doyle, J.C., 1998. Essentials of Robust Control. Prentice Hall, Upper SaddleRiver, NJ.