Active optimal loop control to reduce the seismic response of a nonlinear isolation

Many systems are characterized by the fact that non-linear behaviour is restricted to certain points, particularly in the soil-interaction cases. The structure considered as a single degree of freedom is nonlinearly attached in a singular point, practically simulating the structure soil-interaction. In the first part of the paper we present the general problem for the situation when the dimensions of the state space and of the control are the same.Simple computations using Lie brackets of vector fields show that the optimal feedback law satisfies a system of quasi-linear first order partial differential equations. When the control does not appear in the criterion, this system degenerates into algebraic equations. For the polynomial nonlinearity and quadratic performance index we obtain by the original procedure the optimal law in the analytical form. The theoretical study is applied for to minimize the displacement of an idealized reactor building which is subject to horizontal earthquake-induced accelerations. The active control scheme developed uses a system of force actuators (most probable hydraulic) to counter the action of forcing input due to earthquake. The design problem concerns the optimal control input so that the system responds favourably. The paper is in a certain meaning an extension for the nonlinear optimal cases,soil-structure interaction,of the results of ( Wolf, Madden 1981 ). As a practical examples we present the behaviour of a structure of shear beam linear type with a cubic nonlinearity confined to the connection between the foundation and the moving base.