Structural Damage Identification Based on the Minimum System Realization and Sensitivity Analysis

On the basis of the thought that the minimum system realization plays the role as a coagulator of structural information and contains abundant information on the structure, this paper proposes a new method, which combines minimum system realization and sensitivity analysis, for structural damage detection. The structural damage detection procedure consists of three steps: (1) identifying the minimum system realization matrixes A, B, and R using the structural response data; (2) defining the mode vector, which is based onminimum system realizationmatrix, by introducing the concept of the measurement; (3) identifying the location and severity of the damage step by step by continuously rotating themode vector.The proposedmethod was verified through a fivefloor frame model. As demonstrated by numerical simulation, the proposed method based on the combination of the minimum realization system and sensitivity analysis is effective for the damage detection of frame structure. This method not only can detect the damage and quantify the damage severity, but also is not sensitive to the noise.


Introduction
As our economy enjoys a rapid growth and engineering construction technologies become mature increasingly, the number of high-rise buildings and long-span structures has been increasing.Because all of these structures are landmark of the city where they are located, the damage or collapse will cause great losses of properties and lives.So the performance of these structures under complex environmental factors has attracted extensive attentions.As an effective way of evaluating the safety of a structure, structural health monitoring is employed to track and evaluate damage and deterioration during regular operation and after extreme events for structure [1][2][3][4][5].Although structural health monitoring can be considered as synonymous to damage detection, it actually refers to a much broader research and application area that can be employed for different purposes, such as validation of the properties of new structure, life cycle performance of structure, and structural control.However, damage detection is a very critical component of structural health monitoring.Damage detection in the context of SHM can employ a collection of robust and practical damage detection methodologies to locate and quantify damage of changes in observable behaviors.Over the recent twenty years, a large number of scholars have proposed many methods of damage detection [6][7][8][9][10], for example, damage detection based on the structural dynamic characteristic [11,12].However, in essence, a large part of these methods extract structural damage features from the transformation of the structural dynamic parameters.Mathematically speaking, all of these methods select a mathematical model that can precisely and concisely describe the structure and then achieve the structural dynamic characteristics parameters through specific transformations, so as to detect structural damage.However, this process which gradually condenses extracts and purifies information shows certain disadvantages.On the one hand, repeated mathematical transformation is slightly tedious and would cause many errors in truncation and calculation; on the other hand, the achieved structural mode parameters, for example, frequency, which reflect the whole structural properties, are not sensitive enough to microdamage, early damage, and local damage of the structure; the results of identification based on which the final decision was made will cause many misjudgments.Therefore, it is necessary to explore a more concise method that is easy to operate and sensitive to damages after the relations between structural excitation and response were successfully described by a mathematical model.Therefore, this paper proposes a new method, which combines minimum system realization and sensitivity analysis, for structural damage detection.First, the theoretical background and basis of the minimum system realization are briefly discussed and the minimum structural system realization matrixes A, B, and R of the structure were obtained.Then, the concept of measurement is introduced and the mode vector which characterizes structural information based on the minimum system realization matrix was defined, and then the sensitivity theory was applied to detect structural damage step by step by continuously rotating the mode vector.Finally, the proposed method was verified by a five-floor frame model.

Minimum System Realization
The so-called realization is to determine state space description which described internal structural characteristics system through the transfer function matrix which described external causal relationship of system [13].Given transfer function matrix G() of linear time-invariant system, if [A, B, R] can be obtained to make G() = R(I − A) −1 B tenable, [A, B, R] can be regarded as a realization of the given transfer function matrix G(), namely, a realization of the system.There are various forms of realizing the same system, and transfer function matrix can only reflect the part of the system that can be controlled and observed, and G() can derive the realization of different dimensionalities.These different equations demonstrate different physical structures of the system, among which the minimum dimensionality realization can obtain equivalent external transfer characteristics with the simplest system model.And that is the socalled minimum system realization [14].
A finite dimensional, discrete-time, linear, time-invariant dynamical system has the state variable equations [15]: where x() is an -dimensional state vector; y() is response vector of actual measurement; A, B, and R are system matrix, control matrix, and measurement matrix, respectively; u() is input vector;  is sample indicator.
then the discrete time impulse response function matrix of the structure can be achieved: where where ℎ  () is impulse response function between excitation point  and response point  at time .Thus, the general Hankel matrix can be constructed through the impulse response function matrix h(): where  and  are random positive integers.When  = 0, conduct singular value decomposition (SVD) on H(), namely, H(0); then the following is tenable: where P and Q are vector matrixes of the right and the left singular value and Σ is diagonal matrix shown below:

Measurement
In mathematics, measurement (or distance function) is a function defining the distance between elements in the set.
A set with measurement is called metric space.At the beginning of the 20th century, Fréchet, a French mathematician, found out that many analysis achievements involve the distance relations among functions from a more abstract view, through which the concept of metric space can be abstracted [16].
After achieving the minimum system realization, Z can be defined by A, B, and R: With regard to the health structure and structure which is to be identified, Z can be expressed by In fact, it can be seen that Z 0 , Z  can be rewritten as Z 0 = Z 0 I, Z  = Z  I, where I is unit matrix.Since any matrix can be seen as a coordination system, Z 0 and Z  were described by Z 0 and Z  in the coordinate system I.According to this, the coordinate system Z 0 is employed to describe Z 0 and Z  , so as to obtain D 0 and D  : D 0 and D  are a kind of measurement of Z 0 and Z  in the coordinate system Z 0 .The physical mean of this description in the structure damage detection is to use the coordinate system (Z 0 ) which represents the health states to measure the states to be identified (Z  ); in other words, it explores and quantizes the differences between the states to be identified (Z  ) and the health states (Z 0 ), so as to identify structural damages.
In the theory of matrix, any matrix can be regarded as a linear space, and any row vector of the matrix can be regarded as a linear transformation; a linear transformation in mathematics is also -dimension array [17].With the purpose of describing the array more concisely, a measurement method called 2-norm is brought in.Norm is the function with the concept of "length, " while the norm of vector is used to measure the length of vector.Therefore, norms can be used to describe each row vector of matrix.Then, 2-norms of every row vector in matrixes D 0 and D  are achieved, respectively, and the corresponding mode vectors u 0 and u  are also obtained as follows: Vector u 0 represents essential information of health structure, and vector u  represents essential information of damaged structure.Thereby, after bringing in the concept of measurement, the minimum realization matrix can be successively transformed into mode vector that can basically represent structural information, which can be used for our follow-up damage identification.

Sensitivity Theory
Sensitivity theory is mainly used to quantitatively describe the variances of structural behavior with structural parameters.
In other words, structural parameter  shows a slight variance , how the structural behavior (mode vector u) will vary and in which degree it will vary [18,19].
Since mode vector u is the function of structural parameter , variance  of the structural parameters is also reflected in vector u correspondingly.First-order variation of the structural parameter  with vector u was calculated, and then ( 13) is tenable: where u is variance of mode vector,   is variance of structural parameter in floor , u/  is the sensitivity coefficient of mode vector u of structural parameter   , and  is the number of structural parameters.An -floor structure was taken as an example, and the structural parameter   in floor  was assumed to be Δ  variance; the variance Δu of mode vector u after comparing with the health structure is shown below: where vector u  = [ 1 ,  2 , . . .,   ]  .The left and the right ends of ( 14) were divided by the structural damage degree Δ  , and then we can obtain the variance of structural unit damage information in the floor .Consider the following: As a matter of convenience, s  and   are used to represent Δu/Δ  and Δ  /Δ  ( = 1, 2, . . ., ), respectively, and Mathematical Problems in Engineering then the sensitivity vector s  of mode vector u of structural parameter  in floor  can be achieved: Similarly, we can obtain the sensitivity vectors s 1 , s 2 , . . ., s  of mode vector u to structural parameter  of each floor, respectively.By combining the  vectors, we can get the sensitivity matrix S: A slight variance p of the structural parameter  represents structural damage.Thereby, mode vector u  of damage structure contains the damage information of the structure, and, compared with that of the health structure, the following is tenable: Combining ( 13), the following is tenable: Namely, In other words, as long as the variance u of sensitivity matrix S and vector u is known, the variance of structural parameter can be obtained through (20):

Damage Identification
General sensitivity analysis is shown in (20).The accuracy of the identification result mainly relies on the accuracy of sensitivity matrix.However, the accuracy of sensitivity matrix is not always guaranteed under the influence of various complex factors.Therefore, the accuracy of such identification that "set the tune with one beat of the gong" cannot always be guaranteed.This paper proposes an identification method which continuously modifies the sensitivity matrix to approach actual damage by keeping rotation of the mode vector.The identification process is showed in Figure 1.In order to illustrate the process of the identification, a five-floor frame structure is taken as an example.State 1-State 5 represent the states when a certain severity of damage happens in different locations of the structure.The vectors which were described by Mode 1-Mode 5 represent the process from health state to State 1-State 5, respectively.The direction and characteristics of these vectors represent the location of damage, and the length of vector represents the severity of damage.When an unknown mode vector which points from the health state to an unknown state completely coincides with a known mode (the same in characteristics, direction, and size), single damage of the structure can be identified.If the unknown mode vector is always between two known modes, multiple damages of the structure can be identified by projecting the unknown mode vector to the two modes.
The damage is simulated by reducing the stiffness of the structure; it is assumed that damage  happens to different locations of the structure, respectively (define  as the standard damage value which was adopted to construct sensitivity matrix); five corresponding mode vectors were obtained using the above mentioned method and employed to construct the sensitivity matrix and then to identify unknown damage.Firstly, the similarity between the unknown and known modes can be basically confirmed by comparing the characteristics of unknown and known modes.Higher similarity demonstrates higher possibility that the unknown damage is similar to this standard damage.Thus, the location of structural damage can be judged.Secondly, project the unknown mode vector to the known mode vector after confirming the possible location of damage.The size of projection is the severity of possible damage.Define the damage severity as   which has been located with the maximum possibility.Considering that   is closer to the actual damage severity than , so   is utilized to reconstruct a sensitivity matrix to identify the unknown damage, and   is achieved; similarly,   was utilized to identify again; the iterations stop until the result of  and  + 1 iteration no longer varies.Thus, the final damage is identified.Therefore, it is possible  to identify the location and severity of structural damage accurately by updating the standard damage which was used to continuously modify the sensitivity matrix and rotating the mode vector.

Numerical Example
In order to verify the feasibility and effectiveness of the proposed method in identifying structural damage, a five-floor frame is used in the numerical study.The frame model is showed in and frequency spectrum curve were showed in Figure 3. On the basis of sampling theorem, the responses are recorded at a sampling rate of 50 Hz and lasting for 100 s.A total of 5,000 time points of structural responses are obtained from each floor for the undamaged structure as well as for the structure in damage case.Six damage cases in numerical simulation were given in the Table 1.The damage was simulated by the reduction of the stiffness of the structure, for example, "damage of 20%" represents the stiffness of the structure reduce by 20%.

Selection of Standard Damage.
In order to discuss the influence of the sensitivity matrix constructed by different standard damage on the result of identification, standard damage severities at 5%, 10%, 15%, and 20% were adopted to construct the sensitivity matrix, and the damage cases mentioned above were analyzed.The results can be seen in Figures 4 to 9.
It can be found that this method can locate the damage when the different severity of damage happened in a different floor accurately (floor 1, floor 3 damage) using different sensitivity matrix by comparing Figures 4 and 6, Figures 5  and 7. Comparing Figures 4 and 5, Figures 6 and 7, it can be found that this method can quantify the severity of the damage accurately.As these six figures are further analyzed, it is suggested by Figures 4 and 6 that the standard damage of 5% shows the optimal performance in identifying 3% damage.From Figures 5 and 7 it is suggested that the sensitivity matrix which was constructed by standard damage of 20% shows the optimal performance in identifying 20% damage; it also can be found by Figure 8, Figure 9 that the sensitivity matrix which constructed by standard damage of 10% shows the optimal performance in identifying 12% damage and the sensitivity matrix which was constructed by standard damage  of 20% shows the optimal performance in identifying 25% damage.Thus, it is not difficult to find out that the closer the standard damage value adopted to construct the sensitivity matrix is, the better the identified results are obtained.However, in general, it can be seen from Figures 4 to 9 that the standard damages values selected to construct the sensitivity  matrix are quite different, but they can identify damages in different locations and severity very well.This demonstrates that the selection of standard damage value does not affect the result of damage identification.That is to say that this method has no special requirements to standard damage value.Therefore, it is feasible to approach the actual damage by continuously updating the standard damage value which was used to construct the sensitivity matrix and rotating the mode vectors.

Single Damage Analysis.
Based on the above discussions, the standard damage of 10% was employed to construct the sensitivity matrix.Given that there is no noise in the response data, five iterations are adopted to identify the above mentioned cases.The results were showed in Figures 10-15.
It can be seen from Figures 10-15 that this method can locate the damage and quantify the damage severity under different damage cases accurately.Misjudgments might occur in the first iteration but they will disappear after several iterations.In these figures, Figures 12 and 13 are typical "step approaching, " which approaches the actual damage by increasing or decreasing step by step.Figure 15 is typical "pendulum approaching, " which approaches the actual damage by oscillating near the actual damage.It can be seen from Figures 10 and 12 that this method can identify slight damage, for example, damage of 3%, through two iterations; it can be seen from Figures 11,13, and 15 that despite the serious damages like 21% and 25%, this method can satisfy the requirement of accuracy by three or four iterations.If the selected standard damage value happens to be very close to the actual damage (shown in Figure 14), the result of one iteration is already quite accurate.It is worth noting that there is no misjudgment in several iterations when identifying slight damages.This means that this method shows great advantages in identifying slight damages.Overall, this method takes only a few iterations to identify structural damages, which substantially shortens the time needed to identify the damage.

Antinoise Performance Analysis.
In order to discuss the antinoise performance of this method, white Gaussian noise was added to the response data at the noise-to-signal ratio of 0%, 1%, 3%, and 5%, respectively.The standard damage of  10% in each floor of the structure was employed to construct the sensitivity matrix and the above mentioned cases were analyzed.The results were showed in Figures 16,17,18,19,20,and 21.It is suggested from Figures 16 to 21 that this method shows certain antinoise performance.The location and severity of structural damage can still be identified very well when 5% noise was added to the response of the damage of 21% to 25%; but the noise produces greater influence on slight damage of 3%.Taking case 1 and case 3 as an example, the location and severity of damage can be identified well when 1% and 3% noises were added to the response of the damage structure.When 5% noise was added, misjudgments show up, but they can also locate structural damage.As to case 5, it is indicated by Figure 20 that, when the standard damage and actual damage selected to construct the sensitivity matrix are very close, the influence of noise can be weakened and dramatic fluctuations can be avoided.

Multiple-Damage Analysis.
In practical engineering, multiple structural damages might occur in different locations.This paper also discusses the method in the identification of multiple damages.The five-floor frame shown in Figure 2 is simulated.The standard damages of 8% and 15% were used to construct sensitivity matrix.Given that there is no noise pollution in the response data, five iterations were adopted to identify the different damage cases showed in Table 2.The results were showed in Figures 22 to 25  It is suggested by Figures 22 to 25 that this method can locate structural damage very accurately even with multiple damages, but the severity of damage is not very ideal.It is not difficult to find out by comparing Figures 22 and 23, Figures 24 and 25 that the standard damage selected to construct the sensitivity matrix barely has any influence on the results of multiple damages.It is also found out by Figures 22 to 25 that this method shows satisfying effect when identifying multiple damages consisting of middle and slight damages, but it seems to be helpless in front of multiple damages consisting of middle and large damages because the corresponding mode vector under multiple structural damages is not just a simple superposition of the corresponding mode vector in case of single damages and the coupling among multiple damages has greater influence on the mode vector needed in the identification process.

Conclusion
Based on the minimum system realization and sensitivity analysis, this paper proposed a new method of identifying structural damages.The feasibility and effectiveness of this method was verified by numerical simulation analysis.The main conclusions are as follows.(1) This method has no special requirement to the standard damage selected to construct sensitivity matrix, and this offers great convenience to identify the damage.
(2) This method needs only a few iterations to identify structural damages very accurately through "step approaching" or "pendulum approaching." This greatly shortens the time needed to identify damages.What is more important is that this method shows more remarkable performance in the identification of slight damages.(3) This method shows certain antinoise performance.Even with stronger noise, it can still identify the location and severity of large structural damages; as to slight damages, the influence of noise is large, but this method can still locate the structural damages.
(4) Although this method shows satisfying effect in identifying multiple damages consisting of middle and slight damages, it can only locate middle and large damages because of the coupling among multiple damages.Therefore, further study needs to be conducted in this aspect.(5) Although the proposed damage identification methodology has shown promise in the numerical investigation, issues to be addressed still exist to make it applicable in practice.Therefore, experimental study is needed in further work to investigate the feasibility of the methodology in practical applications.

5 U n k n o w n m o d e M o d e 2 ?Figure 1 :
Figure 1: Process of structural damage identification.

Figure 3 :
Figure 3: Time history and frequency spectrum curve of white noise excitation.

Table 1 :
Single damage case of five-floor frame model.

Table 2 :
Multiple-damage case of five-floor frame model.