Analysis and control for transient responses of seismic-excited hysteretic structures

• Seismic transient response of Duhem hysteretic system to time-modulated excitation.
• Establish the semi-analytical solution of transient response.
• Derive the optimal bounded control strategy to minimize transient response.

The semi-analytical solution of transient responses and the bounded control strategy to minimize the transient responses for seismic-excited hysteretic structures are investigated in this manuscript, the hysteretic behavior is described by Duhem model while the seismic excitations by random processes with Kanai–Tajimi spectrum. The averaged Fokker–Planck–Kolmogorov equation with respect to the probability density of amplitude response is firstly derived by utilizing the stochastic averaging technique based on the generalized harmonic functions. The probability density is approximately expressed as a series expansion in terms of a set of specified basis functions with time-dependent coefficients which are determined through the Galerkin procedure. The quasi-optimal bounded control strategy to minimize the transient response is proposed based on the averaged system with respect to amplitude response and an appropriate performance index. The quasi-optimal control is derived from the minimum condition in the dynamic programming equation. The application and effectiveness of the proposed analytical procedure and control strategy are illustrated through one representative example.