Optimal polynomial control of seismically excited linear structures

In this paper, an optimal nonlinear control law that is polynomial in terms of the states of the system is proposed for limiting the peak response of seismic-excited buildings. A performance index that is quadratic in control and polynomial of an arbitrary order of the states is considered. The performance index is minimized based on the solution of the Hamilton-Jacobi-Bellman equation. The resulting controller is a summation of polynomials of different orders, i.e., linear, cubic,; and quintic, among others. Gain matrices for different parts of the controller are calculated easily by solving matrix Riccati and Lyapunov equations. Extensive simulation results indicate that the new optimal polynomial controller consumes less energy in reducing the peak response quantities; however, it may use bigger peak control force than the linear controller. When the earthquake intensity exceeds the design one, the optimal polynomial controller is capable of exerting larger control forces thus achieving a higher reduction for the peak structural response. Such response adaptive properties are very desirable for the protection of the integrity of civil engineering structures because of the inherent stochastic nature of the peak ground acceleration.