In this section, a method is presented to diagnose the condition of a bridge during a brief interruption in traffic flow by using the acceleration response of the bridge superstructure induced by the action of a moving vehicle traveling across the bridge at constant speed. A condition diagnosis feature is generated by using the innovation between the measured acceleration and the predicted acceleration obtained from the Kalman filter, and then a condition diagnosis index—which is the energy ratio between the innovation and the measured acceleration—is proposed by calculating the null space of the Hankel matrix consisting of condition diagnosis features. Following the basic novel detection idea, the established condition diagnosis index is utilized to assess the condition of the bridge.

#### 2.1. Condition Diagnosis Feature Based on the Innovation Obtained by the Kalman Filter

A bridge is deemed a linear structural system. Thus, on the basis of state space theory, the state equation of a linear discrete system of a bridge is defined as follows:

where

k represents the sampling time point of the structural response (

$k=1,2,\cdots ,N$), in which

N is the count of sampling time points;

${x}_{k}\in {R}^{n\times 1}$ represents the state vector at the

kth sampling time point, wherein

n is the system order, which is twice the total number of degrees of freedom of bridge;

${x}_{k+1}\in {R}^{n\times 1}$ is the state vector at the (

k+1)th sampling time point;

$A\in {R}^{n\times n}$ is the system matrix;

${y}_{k}\in {R}^{m\times 1}$ is the output vector of the structure at the

kth time point;

m is the total number of measured degrees of freedom;

$C\in {R}^{m\times n}$ is the state output matrix;

${v}_{k}\in {R}^{m\times 1}$ is the measured noise vector at the

kth time point; and

${w}_{k}\in {R}^{n\times 1}$ is the noise of the excitation load. The noise of the excitation load is defined by the following equation:

where

${\rho}_{k}\in {R}^{r\times 1}$ is the unmeasured excitation vector at the

kth time point,

r is the total number of excitation loads, and

$G\in {R}^{n\times r}$ is the transfer matrix between the input load and state vector of the system. The vector

${v}_{k}$ is defined by the following equation:

where

$D\in {R}^{m\times r}$ is the transmission matrix and

${\eta}_{k}\in {R}^{m\times 1}$ is the pure measured noise at the

kth time point.

The innovation is defined as the optimal differences between the measured and predicted responses of the bridge obtained by the Kalman filter [

24]. Based on the abovementioned equations, the innovation

${e}_{k}$ is computed as follows:

where

${\widehat{x}}_{k}^{-}$ is the prior state estimation vector at the

kth time point. The posterior state estimation vector at the

kth time point can be estimated by using the Kalman filter [

36], defined as follows:

where

$\overline{K}$ is the steady state Kalman gain, which is computed by using the following equation:

where the covariance matrix of the state error

$P\in {R}^{n\times n}$ is defined as follows:

As described above, the innovation obtained by the Kalman filter is related to the measured responses of bridges, and the measured responses are directly determined by the excitation load acting on the bridge. For a bridge without any damage, the Kalman filter is obtained by using the measured acceleration of the bridge superstructure under the action of a certain excitation load, and the innovation $e$ is calculated by using the generated Kalman filter. If the same excitation load acts on this bridge, another innovation ${e}^{\prime}$ is acquired by using the abovementioned generated Kalman filter. If the structural condition of this bridge does not change, the two innovations, $e$ and ${e}^{\prime}$, should be the same in theory. Therefore, the innovations obtained from the same excitation load can be used to diagnosis the condition of the bridge.

However, for a bridge in operation, it is impossible to keep the excitation load consistent for different times. To solve this issue, a reasonable method is to stop the traffic and to excite the bridge by using the same standard loading vehicle traveling along the same driving route at the same constant speed for every load test. This excitation method is similar to the regular load test of a bridge, and it is easy to implement for real situations. In contrast to the regular load test, we do not need to stop the traffic flow for many hours for each load test because for the selected medium- and small-span beam bridges, the loading vehicle does not require much time to travel across the bridge even though the vehicle speed is very low. As discussed above, the innovations obtained by the Kalman filter using the acceleration responses of the bridge are defined as the condition diagnosis feature. These innovations form the following matrix:

where

j (

$j=1,2,\cdots ,m$) is the total number of measured acceleration responses of the bridge.

#### 2.2. Condition Diagnosis Index Based on the Energy Ratio between the Innovation and the Measured Response

The energy ratio calculated by using the measured acceleration response and its corresponding innovation is defined by the following equation:

where

${\beta}_{j}$ is the energy ratio between the innovation and the acceleration response at the

jth measured point and

${y}_{j}=\left\{\begin{array}{cccccc}{y}_{1,j}& {y}_{2,j}& \cdots & {y}_{k,j}& \cdots & {y}_{N,j}\end{array}\right\}$ is the acceleration response at the

jth measured point. Using Equation (11), the energy ratio of all the measured points can be obtained, and then the following vector is formed:

where

$\mathrm{sort}\left(\cdot \right)$ is the operator of arranging the order from small to large. The vector

$\beta $ contains all the information from the innovations of the predicted acceleration responses. The change in

$\beta $ is the key to diagnosing the variation in the structural condition of a bridge. To obtain consistent and comparable results every time, the vector

$\beta $ should be reordered. In this study, it is recommended to arrange the vector

$\beta $ so that the values change from small to large. After acquiring the vector

$\beta $, the following Hankel matrix, denoted as

**H**, can be formed:

where

$p$ and

$q$ are the number of rows and columns in the Hankel matrix (

$p<q$), respectively.

The elements on the antidiagonal of the Hankel matrix are the same, i.e., two adjacent columns are misaligned. Therefore, any column of the Hankel matrix has a significant correlation with each other, which can be selected as a delay vector as follows:

According to Equation (13), the length of the delay vector is $p$, and the number of delay vectors is $q$. Because adjacent delay vectors are highly correlated, when $p$ is very small, $q$ increases, and the correlation between any two delay vectors decreases. In contrast, if $p$ is large, $q$ decreases, and the correlation between each delay vector increases.

In this study, the first load test of a bridge is defined as the reference condition, and the corresponding Hankel matrix under the reference state is named

${H}_{}^{0}$. The following equation is obtained:

where

${N}_{}^{0}\in {\mathbb{R}}^{q\times 1}$ is any column vector of the right null space of matrix

${H}_{}^{0}$ in the reference state. This column vector is defined as follows:

where

$\mathrm{null}\left(\cdot \right)$ is the operator of calculating the right null space of the matrix and

$\mathrm{column}\left(\cdot \right)$ is the operator of taking any column of one matrix.

In addition to the reference condition, the load test is repeated

${m}_{\mathrm{H}}$ times under the condition of bridge without any structural damage, and we intuitively define these tests as the load tests under the healthy condition of bridge. For the bridge without any damage, the Hankel matrix obtained by using the

${k}_{\mathrm{H}}$th test is denoted as

${H}_{{k}_{\mathrm{H}}}$. Using the generated null space

${N}_{}^{0}$ under the reference condition, the following residual

${\alpha}_{{k}_{\mathrm{H}}}$ (

${\alpha}_{{k}_{\mathrm{H}}}\in {\mathbb{R}}^{p\times 1}$) is obtained from the following equation:

where

${\alpha}_{{k}_{\mathrm{H}}}$ is the vector of residuals under the healthy condition of the bridge and

${k}_{\mathrm{H}}\in \left(1,2,\cdots ,{m}_{\mathrm{H}}\right)$ is the number of load tests.

Theoretically, for the reference condition and the healthy condition defined above,

${H}_{{k}_{\mathrm{H}}}$ is the same as

${H}_{}^{0}$; thus,

${\alpha}_{{k}_{\mathrm{H}}}$ is a perfect zero vector. However, owing to measurement noise, the residual values cannot be zero but are close to zero. Because the number of repeated load tests is small and no statistical characteristics are available, the condition diagnosis index is defined to evaluate the vector of residual. This index is expressed as follows:

where

${\gamma}_{{k}_{\mathrm{H}}}$ is the condition diagnosis index of the

${k}_{\mathrm{H}}$th load test. After obtaining all the condition diagnosis indexes of

${m}_{\mathrm{H}}$ loading tests under the healthy condition of the bridge, the following threshold under the healthy condition of the bridge is defined:

where

$\eta $ is the threshold under the healthy condition of the bridge and

$\theta $ is the guarantee coefficient. Usually, the value of guarantee coefficient should be determined case by case. This value depends on the test condition, the ratio of signal to noise of measured data, pavement situation etc. For the example of actual bridge described in

Section 4, the guarantee coefficient is taken 1.2 according to the real condition of load test.

Except for the abovementioned two conditions, all the other conditions of bridge are defined as the condition to be diagnosed. The residual of the

zth loading test is calculated by using the following equation:

where

${\alpha}_{z}$ is the vector of residuals for the condition to be diagnosed and

z is the number of load tests. The condition diagnosis index for the condition to be diagnosed is calculated by using the following equation:

where

${{\gamma}^{\prime}}_{z}$ is the condition diagnosis index of the

zth load test. If

${{\gamma}^{\prime}}_{z}$ is larger than

$\eta $, the condition of the bridge is deemed abnormal. Conversely, the bridge is considered healthy if

${{\gamma}^{\prime}}_{z}$ is smaller than

$\eta $.

For a bridge in actual operation, especially for newly constructed bridges, it is appropriate to regularly perform the proposed method to diagnose the condition of bridges. With the accumulation of load test data, the statistical characteristics of condition diagnosis indexes can be obtained. Thus, the threshold for the healthy condition of the bridge can be calculated by using a statistical approach. In this way, the robust performance of the proposed method is enhanced. For the proposed method, the results of condition diagnosis do not need to consider the influence of environmental temperature because the time required for a load test is about half an hour on average. However, the environmental temperature of load test for different time should be similar, so the results obtained by different load tests could be compared.

As discussed above, the proposed Kalman filter-based method does not need to establish the FEM of the bridge, so the calculation errors caused by the differences between the analytical FEM and the structural performance of the actual bridge are avoided during the bridge condition diagnosis process. Additionally, compared with other types of structural responses, it is easy to measure the acceleration response and to ensure the high accuracy of the data. Therefore, it is relatively convenient for the practical application of proposed method.