Stochastic Optimal Active/Semi-active Control Of Stay Cables

A stochastic optimal control strategy for a slightly sagged cable using support motion in the cable axial direction is first proposed. The nonlinear equation of cable motion in plane is derived and reduced to the equations for the first two modes of the cable vibration by using the Galerkin method. The partially averaged Ito equation for controlled system energy is further derived by applying the stochastic averaging method for quasi non-integrable Hamiltonian systems. The dynamical programming equation for the controlled system energy with a performance index is established by applying the stochastic dynamical programming principle and a stochastic optimal control law is obtained through solving the dynamical programming equation. A bilinear controller by using the direct method of Lyapunov is also introduced and a comparison between the two controllers shows that the proposed stochastic optimal control strategy is superior to the bilinear control strategy in the sense of higher control effectiveness and efficiency. The same procedure is carried out for the controlled motion in-plane and out-of-plane to show the superiority of the stochastic optimal control strategy. The motions in plane can be further reduced to linear equations with active motion control at the boundary. By applying the stochastic averaging method for quasi integrable Hamiltonian systems and the stochastic dynamical programming principle, a stochastic optimal control law is obtained through solving the dynamical programming equation. Extensive parameter studies are carried out for indicating the features of the proposed control strategy. Finally, the stochastic optimal semi-active control for stay cable multi-mode vibration attenuation by using magneto-reheological (MR) damper is developed. The Bingham model for MR damper is used. The force produced by an MR damper is split into passive and active parts. The passive part is combined with structural damping forces into effective damping forces. The stochastic averaging method for quasi integrable Hamiltonian systems and the stochastic dynamic principle are applied and a stochastic optimal semi-active control law is obtained through solving the dynamical programming equation. For controlled modal energies with an index not involving control force, bang-bang control law is obtained without solving dynamical programming equation. A comparison between the two control laws shows that the stochastic optimal semi-active control strategy is superior to the bang-bang control strategy in the sense of higher control effectiveness and efficiency and less chattering.