“Computational Civil Engineering - "Intersections" International Journal

252 F. Paulet-Crainiceanuwhere Q and R = weighting matrices. The prime sign in Equation 5 means thetranspose.Q and R show the relative importance of minimizing the states (structuralresponse) or the actuating forces. These matrices are square. The first is n-dimensional and the second is m-dimensional, where m is the number of theactuators.Minimizing J and considering the unknown a constant matrix P, the next Riccatiequation takes place:1PA − PBR−1 B′ P + A′ P + 2Q = 0(6)2and the control matrix gain is a constant matrix:1K = R−1 B′P (7)2With the established gain matrix, the necessary forces for actuators are calculatedand the corresponding commands are generated for the actuators.A difficult task for applying the classical method shown above is to choose thecoefficients involved by weighting matrices Q and R. For large systems it is notpossible to accept individual trial for each coefficient. Therefore general rules arebetter to envision.The first term in the brackets of Equation 5 can be written as energy expression andthus leading to minimization of the energy of the structural response. A solution isgot if the weighting matrix Q is composed from the structural matrices M and K.⎡KQ = ⎢⎣ 00 ⎤M⎥⎦A problem occurring when using Equation 8 is that of the embedded active devices'dynamics. It must be noted that the minimization from Equation 5 should refer onlyto the energy of structural response. Therefore, it is advisable to not use the true Mand K matrices but matrices in which the terms provided by the active devices areconsiderably diminished.In what is regarding the weighting matrix R it can be taken as a diagonal matrixwith terms showing the relative importance between the active devices.(8)