13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- To minimize the least-squares error Ein Eq.(6) over the sequence of the forcing vector, the dynamic programming method and Bellman's Principle of Optimality are applied. This leads to defining the minimize value of Efor any initial xand the number of stages, n. Thus: ( ) min ( , ) i n n i f F x E x f (7) The recurrence formula can be derived by applying the Principle of Optimality: 1 1 1 1 1 1 1 1 2 1 1 1 ˆ ˆ ( ) min[( ) ( ) ( ) ( ) ( )] n T T n n n n n n n n n n f F x Qx y Qx y f f F Cx Df (8) This equation represents the classic dynamic programming structure in that the minimizing at any point is determined by selecting the decision 1 nf to minimize the immediate cost (first and second terms) and the remaining cost resulting from the decision (the third term). The solution is obtained by starting at the end of the process, n N , and working backward toward 1 n . At the end point n N , the minimum is determined from: 1 2 ˆ ˆ ( ) min[( ) ( ) ( ) ( )] N T T N N N N N N N f F x Qx y Qx y f f (9) At this end point the minimum is obtained by choosing 0 Nf which gives: 1 ˆ ˆ ( ) min[( ) ( )] N T N N N N N f F x Qx y Qx y (10) Eq.(10) can be expanded to: 1 1 1 ˆ ˆ ˆ ( ) ( , ) 2( , ) ( , ) T T N N N N N N N F x x Q Qx x Q y y y (11) Eq.(11) can be changed as: ( ) ( , ) ( , ) N N N N N N N F x x R x x S q (12) where 1 T NR Q Q , 1 ˆ 2 T N N S Q y , 1 ˆ ˆ ( , ) N N N q y y . Eq.(12) shows that NF is quadratic in Nx . It can be proven inductively that all of the nF are quadratic in nx , thus for anynwe can write: ( ) ( , ) ( , ) n n n n n n n F x x R x x S q (13) Substituting Eq.(13) into Eq.(8) and minimizing the equation, the optimal forcing term * 1 n f : * 2 1 1 (2 2 ) 2 T T T n n n n n D R D f D S D RCx (14) For simplification the Eq.(14), let： 1 2 (2 2 ) T n n V D R D (15) 2 T n n H D R (16) Eq.(23) can now be written as: * 1 1 T n n n n n n f V D S V H Cx (17) These are recurrence formulas required to determine the optimal solution of Eq. (6). Using Pearson product-moment correlation coefficient to measure the relationship between identification results and actual results, usually expressed by . The equation can be expressed by:
RkJQdWJsaXNoZXIy MjM0NDE=