Structural damage mitigation in discrete structures using multiple actuators

1. Proceedings of the International Conference on Emerging Trends in Engineering and Management (ICETEM14)
30 – 31, December 2014, Ernakulam, India
165
STRUCTURAL DAMAGE MITIGATION IN DISCRETE
STRUCTURES USING MULTIPLE ACTUATORS
Manmathakrishnan P1
, Athibaranan S2
, Lakshmanan D3
1
(Aeronautical Engineering, BIT, Sathyamangalam, India)
2
(Civil Engineering, BIT, Sathyamangalam, India)
3
(Aeronautical Engineering, BIT, Sathyamangalam, India)
ABSTRACT
Structural Health Monitoring (SHM) is online health monitoring system and enables instantaneous maintenance
triggers when the system’s health falls below predefined level of confidence due to structural damage. The inference is
that Current SHM techniques could not exclude the possibilities of a structural failure to occur in real-time, Active self-
healing mechanism is a solution technique. Where the four levels of SHM is pursued based on the vibration database in
real-time, once damage is detected if it grows structural damage mitigation is applied through actuator technologies,
which is the Vibration control in damaged structures and also called Active structural prognosis, Here damage growth
mitigation using state and state derivative feedback controller is illustrated for multiple input systems. Then to illustrate
its utility, structural damage growth mitigation (active structural prognosis) using time response shaping under a
structural load is illustrated. It is shown that the life of the damaged structure in a dynamic environment is possible to
extend using time response options resulting from a typical state/state derivative feedback controller. A discrete
dynamic structure is considered to illustrate the underlying principles of active structural prognosis.
Keywords: Active Structural Prognosis, Damage Mitigation, Multiple Input Systems, State Derivative Feedback
Controller, Structural Health Monitoring.
I. INTRODUCTION
Structural Health Monitoring (SHM) has become a useful tool for detecting when the characteristics of a
structure have changed to indicate damage such that well-timed and effective maintenance may be planned and the
remaining performance capacity may be assessed. SHM has also lead to a better understanding of the loads and the
response with in a structure in order to optimize design. Substantial developments have taken place in the areas of
structural health monitoring, wherein, the objective is not only to detect damage but also to determine its size and
location across the structure. Often, the damaged structures do not become obsolete unless the damage is indeed severe.
Thus, it is important to mitigate the growth of damage and prevent the possibilities of a structural failure. Vibration is an
important source of damage growth. Therefore, it is important to control vibrations such that the damage growth is
mitigated. Suppose an effect of damage in the structure is said to be mitigated when the vibration response of the
damaged structure with control is same as the vibration response of the undamaged structure. Then one of the
requirements is that the closed loop eigenstructure of the damaged structure needs to match with the open loop
eigenstructure of the undamaged structure. In this framework, this paper uses a linear algebraic technique to assign the
undamaged eigenvalues and eigenvectors for the damaged structure with a stiffness loss using a state and state derivative
feedback controller. Through this technique, it is shown that the vibration control using an actuator load can assist to
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mitigate damage growth by reducing the vibration response magnitudes at the damaged structural degrees of freedom,
while these magnitudes at other nodes are allowed to increase due to the actuator loads. Vibration control in this sense for
structural damage mitigation is illustrated using discrete structures. From the principles of solving a system of linear
constant coefficient ordinary differential equations available in state-space form, eigenvectors can be shown to modify
the transient time response when eigenvalue locations are all fixed in the open left half plane. In linear systems with
multiple control inputs, procedures to determine state [Andy et al., 1983; Moore 1976] and output [Srinathkumar
1978] feedback controllers to assign eigenvalues and their eigenvectors to shape the transient time response are
well established in the literature. These techniques are referred as eigenstructure assignment techniques. One of the
assumptions made in eigenstructure assignment is that the number of control variables available to determine the
controller is more than one. Secondly, present eigenstructure assignment techniques are not focused to perform time
response modifications due to an external input such as a structural load in an aircraft. These are especially true
in single input systems. In this paper, it is shown that when state derivative feedback is augmented to the existing
state feedback case, both the assumptions of the eigenstructure assignment technique can be relaxed. In this
framework, a multiple input controllable linear system is considered. A novel design procedure to determine
parameterized state and state derivative feedback controllers under the eigenstructure assignment technique is
presented. To illustrate an acceptable time response due to an external input, a discrete spring- mass-damper
system with a known damage in one of the springs is assumed. Eigenvector options to determine actuator loads that
reduce vibration activity at the damaged spring are illustrated.
The choice of eigenvectors for various stability and performance benefits is well established in the literature
[Liu and Patton 1998; Srinathkumar 2011; Bachelier et al., 2006; Satoh and Sugimoto, 2004]. For instance near
orthogonal eigenvectors are shown to have maximal tolerance to parameter variations at the arbitrary closed loop matrix
entries [Kautsky et al., 1985; Kautsky and Nichols, 1985]. When the eigenvectors are perfectly orthogonal, eigenmodes
get decoupled, which has been a traditional design (performance) objective in aircraft control applications [Sobel and
Shapiro, 1985; Sobel et al., 1994; Nieto-Wire and Sobel, 2007]. When these controllers are implemented in the actual
nonlinear aircraft, flight control modes such as ascend and descend modes are achieved. In fact, eigenvector options are
parameterized and each option is showscon to offer a variety of ascend and descend modes [Ashokkumar, 2012]. These
eigenvectors which are basically aimed to alter transient time responses in a multiple input framework also alters input-
output time responses. These results are available in the disturbance decoupling problems as well as in the fault detection
and isolation algorithms [Shen et al., 1998]. Recently, the parameterized eigenvector options are effectively used in
active self-healing mechanisms for discrete dynamic structures [Ashokkumar, 2013]. Majority of these benefits through
eigenvectors corresponding to their respective eigenvalues fixed in the open left-half plane of the complex plane are
acquired using a feedback controller in state and output feedback formats. An advantage of augmenting a state derivative
feedback controller is that it increases the number of free parameters available to assign the eigenvector elements. This
property is employed in single input systems and time response modifications are performed due to an unknown external
input as in H-infinity control methodology [Satoh and Sugimoto, 2009], where a compensator is designed to
accommodate a norm bounded exogenous input-output relationship. State and state derivative feedback using
eigenstructure options present similar results.
In the areas of applications, structural damage growth mitigation (active structural prognosis) is considered.
Substantial progress has been made in structural health monitoring techniques where the objective is to detect, isolate and
assess the damages in the structures [Sundaresan et al, 2006; Kearns and Ihn, 2009; Williams, 2008; Montalvao et al,
2006; Fritzen, 2005; Doebling et al, 1996]. The damaged structures do not become obsolete unless the damage is indeed
severe. These structures found during its operation are expected to be active until a repairing or a replacement is
performed on ground as in the case of an aircraft or with similar situations in other structures. A procedure to monitor
damages online has been reported in the literature [Ashokkumar, 2013]. One of the options available to extend the life of
the structure in such situations is structural control. In this framework, structural damage growth mitigation problems are
proposed [Ashokkumar and Iyengar 2011 Korkmaz et al, 2011; Korkmaz et al, 2012], where in [Ashokkumar and
Iyengar, 2011] eigenvector options to determine an acceptable transient response under a state feedback control law is
presented. In this paper, similar problem under structural loads is addressed. State and state derivative feedback using
eigenvector options in a single input framework are investigated.
The paper is organized as follows. Time response options in multiple input systems are discussed in Section 2.
Procedures to compute state/state derivative feedback controllers are presented in Section 3. Active structural prognosis
methods are presented in Section 4. Results and discussions and concluding remarks are presented in Sections 5 and 6,
respectively.
II. STRUCTURAL HEALTH MONITORING
Structural Health Monitoring (SHM) is online health monitoring system and enables instantaneous maintenance
triggers when the system’s health falls below predefined level of confidence. An SHM system’s key focus is to monitor
aspects related to damages and load conditions which have direct influence on fitness of structure for service. Multi

3. Proceedings of the International Conference on Emerging Trends in Engineering and Management (ICETEM14)
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167
faceted functionalities of an SHM system include detection of unanticipated structural damage events, damage location
identification, and damage characterization through imaging, monitoring damage growth and enabling feedback action /
alarm mechanism. SHM system makes embedded non-destructive testing sensors an integral part of the structure and
operates with minimal manual intervention. The four key levels of SHM practices are given by
Level 1: Determining whether damage is present in the structure.
Level 2: Identifying the geometric location of damage in the structure.
Level 3: Quantifying the severity of the damage.
Level 4: Predicting the remaining service life of the structure. In this paper remaining service life of the damaged
structure is predicted by structural damage growth mitigation using actuators load.
III. TIME RESPONSE OPTIONS IN MULTIPLE INPUT SYSTEMS
In this section, it is shown that with state and state derivative feedback, eigenstructure assignment for the time
response modifications in single input systems are possible. Consider a single input linear time invariant system of the
form,
x Ax Bu
•
= + (1)
Where x(t) is an n-component state vector. u(t) is the control variable and (A,B) is the controllable pair. The time
(t) derivative of the state vector is represented by the dot. State and state derivative feedback will be of the form,
1 2u Kx G xφ φ
•
= − − (2)
Here iφ take’s the value 0 or 1, but not both of them are zeros. The state and state derivative feedback gains are
represented by k and g, respectively. They are the n-component row vectors. Substituting (2) in equation (1),
2 1(I bG) (A bK)x xφ φ
•
+ = − (3)
Here I is an identity matrix of order n. From the first principle of solving a system of constant coefficient linear
differential equations, choose
0(t) X t
x eλ
= (4)
Equation (4) offers the following generalized eigenvalue-eigenvector constraint,
2 0 1 0(I bG) (A bK)XXφ λ φ+ = − (5)
Hence the eigenvector contributions in the time response is well understood using equations (4) and (5),
respectively. Clearly the eigenvector options to shape time response in multiple input system is illustrated with 1φ =1 and
2φ =1. From equation (5), infer that X0 is,
[ ] 01
0
0
( I A)
GX
X b b
KX
λ λ−  
= − −  
 
(6)
For a given eigenvalue λ in the open left half plane of the complex plane, equation (6) suggests that the basis
vectors spanning X0 are more than one, where GX0 and KX0 are the constants(to be determined) of the basis vectors
spanning X0. When the eigenvalues λ are in the open left half plane of the complex plane, the transient and input-
output responses are given as follows. Consider 1 1φ = and 2 1φ = . The linear time-invariant system with an unknown
exogenous input d(t) is derived as follows.

4. Proceedings of the International Conference on Emerging Trends in Engineering and Management (ICETEM14)
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168
(I bG) (A bK) (t)x x d
•
+ = − + Γ (7)
Here, Γ represents a column vector. Further simplifying equation (7),
1
(I bG) (t)cx A x d
•
−
= + + Γ (8)
Where
1
(I ) (A bK)CA bG −
= + −
Hence the transient response due to a non-zero initial condition x(0) is,
(t) e (0)cA t
ax x= (9)
Input-output response due to exogenous input d(t) is,
(t ) 1
0
(t) (I bG) (t)c
t
A
b
e d dx
τ
τ− −
= + Γ∫ (10)
Clearly, various eigenvector options and state derivative feedback gain G modify the input-output response as in
the case of the H-infinity control [satoh and sugimoto, 2009]. In this framework this paper first presents parameterized
eigenvector options that determine various state and state derivative feedback gains K and G, and hence various time
response options xa(t) and xb(t). Secondly, a suitable time response for structural damage growth mitigation in a linear
spring of a discrete dynamic structure is illustrated.
IV. COMPUTATION OF PARAMETERIZED CONTROLLER GAINS K AND G
A procedure to compute state and state derivative feedback gains k and g, parameterized with respect to the
eigenvector options are presented. One of the properties of linear algebra [Leon, 2002; Strang, 1988] is employed to
compute to compute K and G. When r jjλ λ λ= ± is complex conjugate, the eigenvector X0= r jv jv± . The
generalized eigenvalue-eigenvector constraint in equation (5) becomes,
R R R RA x B y= (11)
Here AR and BR are given by,
( ) 0
;
( ) 0
r j r j
R R
j r J r
I A I B B B
A B
I I A B B B
λ λ λ λ
λ λ λ λ
− − − −   
= =   − −  
(12)
Were xR and yR
r
j
R
r
j
v
v
x
G v
G v
 
 
 =
 
 
  
And
r
R
j
Kv
y
Kv
 
=  
 
(13)
For the desired eigenvalue locations r jjλ λ λ= + , AR and BR are known. Vectors xR and yR are unknowns. The
principle of linear algebra is stated in the following theorem.

5. Proceedings of the International Conference on Emerging Trends in Engineering and Management (ICETEM14)
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Theorem 1 is satisfied if we pick a set of basis vectors δi orthogonal to the columns of BR such that they are also
Orthogonal to ARxR. That is, choose δi in the null space of BR
'
(prime denotes the transpose). Since δi is also orthogonal to
ARxR, we have,
δi
'
ARxR = 0, i=1……Л (14)
Hence Л is the dimension of the null space of BR . Now denote 1[ ..... ]πδ δ δ= . Then equation (14) suggests,
0T
R RA xδ = (15)
By denoting the basis vectors as Kη , this spans the null space of
T
RAδ . Let 1......K l= clearly,
1 1 ....
r
j
R l l
r
j
v
v
x
Gv
Gv
ρη ρη ηρ
 
 
 = = + + =
 
 
  
(16)
Here, 1' .....k lρ ρ ρ= and 1.....k lη η η= . Equation (16) suggests that eigenvectors vr and vj are available as a
function of the free parameters ρ (any numbers not all of them are zeros). That is, depending upon the free parameters
we choose, eigenvector elements vary. Accordingly the scalars Gvr and Gvj are known for each eigenvalue λ . For the
computed eigenvectors, these scalars offer a set of linear algebraic equations with n unknown gain elements. Hence the
gain vector G can be computed whenever the free parameters ρ for each eigenvalues are fixed.
Theorem 2: Given δ in the null space of Rb and RA for a desired distinct eigenvalue set λ , the ρ -parameters
(any numbers not all of them are zeros) spanning the basis vectors for the null space of
T
RAδ uniquely determine the
state/state derivative feedback gains K and G respectively. The state feedback gains K as a function of the free
parameters ρ is computed as follows.
Equation (11) can be written as,
[ ] 0R
R R
y
b A η
ρ
 
− = 
 
(17)
For the controllable pair (A, b), equation (17) in reduced row echelon form [Leon, 2002] gives,
r
R
j
Kv
y R
Kv
ρ
 
= = 
 
(18)
Here R is a known matrix compatible to the ρ -vector. Equation (18) suggests that the scalars Kvr and Kvj for
each eigenvalue λ are determines as in the case of state derivative feedback gain G. Hence the state feedback gain
vector K is computed.
V. STRUCTURAL DAMAGE GROWTH MITIGATION
Consider a typical discrete structure as shown in Figure 1.
Fix the actuator load u(t) at mass 1. Float the structural load d(t) acting at one of the three masses. Assume that
one of the structural health monitoring techniques detected a damage in the spring numbered 3. Hence the objective of
this paper is to mitigate the damage growth in spring 3. Damage is an indication of stiffness loss. As a result, when
pristine and damaged structures are compared, the displacement activity at spring 3 in damaged structure will be more
compared to the displacement activity at other springs. For damage growth mitigation, however, the tensile and

6. Proceedings of the International Conference on Emerging Trends in Engineering and Management (ICETEM14)
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170
compressive activity at the damaged spring must be kept minimal. For the displacement directions in Figure 1, the tensile
and compressive activity at spring 3 is best depicted by an output variable y(t), such that
2 3(t) ( (t) q (t))y q= ± − (19)
This variable y (t) is depicted in Figure 2.
Let, 1 2 3(t) [ (t) (t) (t)]T
q q q q= ,
Then consider the linear system
1 2 1
00 0
u(t) (t)
J
qIq
d
DA A Bqq
          = + +                
(20)
Or, (t)jx Ax bu B d
•
= + + (21)
The objective is to design a state/state derivative feedback control law in equation (2) with 1iφ = , such that φ
is minimized.
(t)
t
yφ = ∑ (22)
Clearly, time response options using parameterized controllers K and G are sought to minimizeφ . In this the
ρ - parameters (free parameters) are used to choose minimum over the parameterized controllers K and G.
VI. RESULTS AND DISCUSSIONS
The mathematical model for the discrete dynamic structure
Shown in Figure 1 is given by [Ashokkumar and Iyengar, 2011],
1
20 10 0
5 15 10
0 6.6667 13.3333
A
− 
 = − 
 − 
2
1.5 1 0
0.5 1 0.5
0 0.3333 0.5
A
− 
 = − 
 − 
1
1 0
0 1
0 0
B
 
 =  
  
1 0 0
0 0.5 0
0 0 0.3333
jD
     
     =      
          
1,2,3.j =
JD Corresponds to disturbance d (t) at mass 1 (D1) or at mass 2 (D2) or at mass 3 (D3), respectively. The eigenvalue
locations are selected as,

7. Proceedings of the International Conference on Emerging Trends in Engineering and Management (ICETEM14)
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-0.977191627000730 + 5.07893785395178i
-0.439696149201781 + 4.11246167441094i
-0.0831122237974887 + 2.08898533745723i
λ
 
 =  
  
The ρ - parameters are arbitrarily selected. Accordingly, the procedure presented in Section 3 leads to the
following state and state derivative feedback controllers. The four controller options are given as follows,
Controller Option 1:
0.185761 0.630072 -2.553277 -3.880598
-0.296573 -0.318880 4.473369 6.157213
-0.428599 -0.763649 -6.768973 -9.840852
[G',K']
-0.297548 -0.427028 -0.146312 -0.388584
-0.143547 -0.240247 0.783270 1.371236
-0.036099 -0.110733
=
-0.263710 -0.780043
 
 
 
 
 
 
 
 
 
1.0e + 002
Controller Option 2:
-1.607514 -6.028431 1.521623 2.158686
-2.205522 -1.034075 2.846625 4.196623
-3.806399 -9.081026 4.459838 4.217093
[K',G']
-7.175908 -1.579464 5.680177 2.638551
-8.560880 -9.970832 6.637884 -9.082105
-4.307556 1.436979 -3.4
=
19689 -6.545565
 
 
 
 
 
 
 
 
 
1.0e + 002
Controller Option 3:
0.670029 1.136555 -2.005601 -3.343263
-0.726383 -0.657687 14.620650 21.138631
-0.629146 -1.005699 9.054600 11.788591
[K',G']
-0.282164 -0.301033 2.173187 2.879823
-0.139662 -0.284300 0.197303 1.088745
0.011246 0.010734 3.1
=
14260 5.081036
 
 
 
 
 
 
 
 
 
1.0e + 002
Controller Option 4:
0.185761 0.630072 -2.553277 -3.880598
-0.296573 -0.318880 4.473369 6.157213
-0.428599 -0.763649 -6.768973 -9.840852
[G',K']
-0.297548 -0.427028 -0.146312 -0.388584
-0.143547 -0.240247 0.783270 1.371236
-0.036099 -0.110733
=
-0.263710 -0.780043
 
 
 
 
 
 
 
 
 
1.0e + 002
A disturbance (resembling an unknown structural load),
d(t) = sin(t) (23)
is assumed active at one of the masses. From the figure 3 comparing all the three cases for four controller options, it is
observed that when disturbance and actuator loads are collocated, least tensile and compressive action at spring 3 is

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172
observed and a single controller will holds good for all the three cases of structural load, which defines the robust control
technology. β-values in Table 1 depict this phenomenon; lower the value in the table indicates the better structural
damage mitigation. For instance, controller option 4 performs in a best possible way compared to other controllers for all
the cases when disturbance floated at any one of the masses and actuator loads are at mass 1 and 2. This indicates that
robust or reliable controller design for damage mitigation. Hence in this paper, time response options using β -values for
structural damage growth mitigation at a linear spring in discrete dynamic structures are presented. β- Parameters are
picked arbitrarily to determine the state/state derivative feedback controller options K and G, respectively.
VI. FIGURES AND TABLES
Fig. 1: Discrete structure for damage growth mitigation in spring with stiffness k3
Fig. 2: Output y(t) with tensile and compressive activity in spring with stiffness k3
Table 1: Time response options for damage growth
Mitigation in spring k3
β- Values Un
controlled
C1 C2 C3 C4
d(t) at mass1 32.4041 5.7088 7.4961 11.1151 3.6217
d(t) at mass 2 61.3676 5.2148 7.0020 10.6211 3.1277
d(t) at mass 3 10.3760 6.0381 7.8254 11.4444 3.9510

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Fig. 3: φ -values for an actuator load at Mass 1 and2, the disturbance at mass 1 ( Case 1), mass 2 ( Case 2), and at
mass3 ( Case 3 ), respectively
VII. CONCLUSION
In the dynamic environment, it is important to mitigate the growth of damage and prevent the possibilities of a
structural failure, which is the main aim of the SHM. Vibration is an important source of damage growth. Therefore, it is
important to control vibrations such that the damage growth is mitigated. A general belief in eigenstructure assignment
techniques is that the transient response modification is possible only if the number of inputs is more than one. In this
paper, it is shown that in multiple input systems, more options for time response (both transient and steady-state)
modifications are possible provided a state derivative feedback is augmented to the existing state feedback cases. In this
framework, a novel procedure based on the principle of linear algebra that computes state/state derivative feedback gains
is presented.
Then as an application problem, a procedure to link structural health monitoring techniques with controls is
considered. The resulting controller will act as robust (or reliable) controller, in which single controller can be used for
all the three cases of structural load. It is shown that the life of an active damaged structure can be extended using the
structural control technology such as the time response shaping options presented in this paper. The resulting controller is
expected to be in place until a replacement or a repairing is performed after the structure comes to a halt.
VIII. ACKNOWLEDGEMENT
The first author gratefully acknowledges Dr Chimpalthradi R Ashokkumar, at Jain University; for his valuable
support.
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