A hybrid Modified Artificial Bee Colony and extended Kalman filter algorithm for structural system identification

Abstract

The continuous monitoring of structural condition is extremely important for sustaining and preserving the service life of civil structures. It provides resolute and staunch information on the health, serviceability, integrity and safety of structures. System Identification (SI) is a process of extracting a “vibration signature” and comparing the present signature to that of the reference (i.e., undamaged) state. Any notable difference in the vibration signature of a system can be attributed to a certain type of degradation. However, developing efficient SI techniques for civil structures is a critical challenge that needs to be addressed. Hence, an effort is made to formulate SI as a non-convex optimization problem. The Modified Artificial Bee Colony (MABC) is assimilated with extended Kalman filter (EKF) for damage identification using modal data. An objective function obtained by the fractional changes of damaged and undamaged structure is controlled by the metaheuristic optimization algorithm embedded in the proposed damage-detection framework. The residual error that arises in modeling is minimized, creating a more accurate prediction for the damage severity and location. The effectiveness of the proposed SI method is demonstrated via numerical simulation and real-time experimental study. The results of the study identify the damage (both location and severity in a single step of optimization) with lesser computational time and faster convergence when compared with Artificial Bee colony (ABC) and Particle Swarm Optimization (PSO) algorithm. This article emphasizes the importance of continuous structural condition monitoring of civil engineering structures.

Appendix

Pseudo-code structure of the SI algorithm

Calculation of stiffness matrix

• 1. Calculate the value of L1, L2, and L3 initially.

• 2. Calculate the value of G and J as

  $$J=\left[\frac{WH\left({W}^{2}+{H}^{2}\right)}{12}\right],$$
  $$G=\left[\frac{W{H}^{2}}{3}+\frac{1.8H}{W}\right].$$

• 3. Determine K1, K2, and K3…K10 values as shown below:

  $$\begin{aligned} k1 & = \frac{EA}{L}; k2 = \frac{GI_x }{L};k3 = \frac{12EI_y }{{L3}} ; k4 = \frac{12EI_z }{{L3}};k5 = \frac{6EI_y }{{L2}};k6 = \frac{6EI_z }{{L2}}; \\ k7 & = \frac{4EI_y }{L};k8 = \frac{4EI_z }{L};k9 = \frac{2EI_y }{L};k10 = \frac{2EI_z }{L}. \\ \end{aligned}$$

• 4. Generate the stiffness matrix as

  $$\left[\begin{array}{cccccccccccc}k1& 0& 0& 0& 0& 0& -k1& 0& 0& 0& 0& 0\\ 0& k4& 0& 0& 0& k6& 0& -k4& 0& 0& 0& k6\\ 0& 0& k3& 0& -k5& 0& 0& 0& -k3& 0& k5& 0\\ 0& 0& 0& k2& 0& 0& 0& 0& 0& -k2& 0& 0\\ 0& 0& -k5& 0& k7& 0& 0& 0& k5& 0& k9& 0\\ 0& k6& 0& 0& 0& k8& 0& -k6& 0& 0& 0& k10\\ -k1& 0& 0& 0& 0& 0& k1& 0& 0& 0& 0& 0\\ 0& -k4& 0& 0& 0& -k6& 0& k4& 0& 0& 0& -k6\\ 0& 0& -k3& 0& k5& 0& 0& 0& k3& 0& k5& 0\\ 0& 0& 0& -k2& 0& 0& 0& 0& 0& k2& 0& 0\\ 0& 0& -k5& 0& k9& 0& 0& 0& k5& 0& k7& 0\\ 0& k6& 0& 0& 0& k10& 0& -k6& 0& 0& 0& k8\end{array}\right].$$

• 5. Check if the member axes coincide with the global axes, if yes then [K] is the elemental stiffness matrix.

• 6. If not establish a relation between the local and global systems by using transformation matrix.

• 7. Determine

  $$r=\left[\begin{array}{ccc}CXx& CYx& CZx\\ CXy& CYy& CZy\\ CXz& CYz& CZz\end{array}\right],$$

  where $CXx=\frac{({x}_{2}-{x}_{1})}{L}$;$CYx=\frac{\left({y}_{2}-{y}_{1}\right)}{L}$; $CZx=\frac{\left({z}_{2}-{z}_{1}\right)}{L}$; $D=sqrt(CXx*CXx+CYx*CYx)$;$CXy=\frac{-CYx}{D}$; $CYy=\frac{CXx}{D}$; $CZy=0$; $CXy=\frac{-CXx*CZx}{D}$; $CYz=\frac{-CYx*CZx}{D}$; $CZz=D.$

• 8. Determine R′.

• 9. End.

• 10. Calculate the elemental equation of space frame element as

  $$KKK=R*KK*{R}^{^{\prime}}.$$

  Calculation of Mass Matrix

  • 1. Calculate the value of L, L2 and rx2.

  • 2. Determine m1, m2, and m3…m10 values as shown below

    m1 = 70; m2 = 35; m3 = 78; m4 = 70*rx2; m5 = 35*rx2; m6 = 27; m7 = 22*L; m8 = 13*L; m9 = 8*L2; m10 = 6*L2;

  • 3. Generate the mass matrix as shown below 

    $$MM = \left[ {\frac{(rho*A*L)}{{105}}} \right]*\left[ {\begin{array}{*{20}c} {m1} & 0 & 0 & 0 & 0 & 0 & {m2} & 0 & 0 & 0 & 0 & 0 \\ 0 & {m3} & 0 & 0 & 0 & {m7} & 0 & {m6} & 0 & 0 & 0 & { - m8} \\ 0 & 0 & {m3} & 0 & { - m7} & 0 & 0 & 0 & {m6} & 0 & {m8} & 0 \\ 0 & 0 & 0 & {m4} & 0 & 0 & 0 & 0 & 0 & { - m5} & 0 & 0 \\ 0 & 0 & 0 & 0 & {m9} & 0 & 0 & 0 & { - m8} & 0 & { - m10} & 0 \\ 0 & 0 & 0 & 0 & 0 & {m9} & 0 & {m8} & 0 & 0 & 0 & { - m10} \\ {m2} & 0 & 0 & 0 & 0 & 0 & {m1} & 0 & 0 & 0 & 0 & 0 \\ 0 & {m6} & 0 & 0 & 0 & {m8} & 0 & {m3} & 0 & 0 & 0 & { - m7} \\ 0 & 0 & {m6} & 0 & { - m8} & 0 & 0 & 0 & {m3} & 0 & {m7} & 0 \\ 0 & 0 & 0 & { - m5} & 0 & 0 & 0 & 0 & 0 & {m4} & 0 & 0 \\ 0 & 0 & {m8} & 0 & {m10} & 0 & 0 & 0 & {m7} & 0 & {m9} & 0 \\ 0 & { - m8} & 0 & 0 & 0 & { - m10} & 0 & { - m7} & 0 & 0 & 0 & {m9} \\ \end{array} } \right]$$

• 11. Check if the member axes coincide with the global axes, if yes then [K] is the elemental mass matrix.

• 12. If not establish a relation between the local and global systems by using transformation matrix and determine the mass matrix.

• 13. Determine R, R′.

• 14. Calculate the elemental equation of space frame element as

  $$MMM = R*MM*R^{\prime}.$$

Determination of the objective function

• 1. Calculate natural frequency ω_i of the ith mode of the degraded frame based on numerical analysis.

• 2. Introduce a reduction factor and calculate the natural frequency of the damaged frame.

$${\text{num}}\_{\text{del}}\_{{w}}\left( {{i}} \right) = \left( {\left( {{\text{abs}}\left( {{\text{num}}\_{\text{fr}}\_{\text{udam}}\left( {{i}} \right) - {\text{num}}\_{\text{fr}}\_{\text{dam}}\left( {{i}} \right)} \right)*{\text{reduction}}\_{\text{alpha}}\left( {{i}} \right)} \right)/{\text{num}}\_{\text{fr}}\_{\text{udam}}\left( {{i}} \right)} \right).$$

• 3. Similarly calculate using FEM analysis as,

  $${\text{fem}}\_{\text{del}}\_{\text{w}}\left( {\text{i}} \right) = \left( {\left( {{\text{abs}}\left( {{\text{fem}}\_{\text{fr}}\_{\text{udam}}\left( {\text{i}} \right) - {\text{fem}}\_{\text{fr}}\_{\text{dam}}\left( {\text{i}} \right)} \right)} \right)/{\text{fem}}\_{\text{fr}}\_{\text{udam}}\left( {\text{i}} \right)} \right).$$

• 4. The fractional changes in natural frequency of the undamaged and damaged frame is given as, fr_fractional_changes = (num_del_w(i)-fem_del_w(i))^2.

• 5. Calculate mode shape of the ith mode of the undamaged frame based on numerical analysis.

• 6. Introduce a reduction factor and calculate the mode shape of the damaged frame. num_del_shape(i) = ((abs(num_shape_undam(i)-num_shape_dam(i))*reduction_alpha(i)) )

• 7. Similarly calculate for FEM analysis as,

  $${\text{fem}}\_{\text{del}}\_{\text{shape}}\left( {\text{i}} \right) = \left( {\left( {{\text{abs}}\left( {{\text{fem}}\_{\text{shape}}\_{\text{udam}}\left( {\text{i}} \right) - {\text{fem}}\_{\text{shape}}\_{\text{dam}}\left( {\text{i}} \right)} \right)} \right)} \right)$$

• 8. The fractional changes in natural frequency of the undamaged and damaged frame is given as, shape_fractional_changes(i) = (num_del_w(i)-fem_del_w(i))^2.

• 9. Determine the objective function as, sum_shape = fr_fractional_changes + shape_fractional_changes(i).