Semi-active optimal control of linearized systems with multi-degree of freedom and application

Abstract

A semi-active optimal control method for non-linear multi-degree-of-freedom systems and its application to a building structure for random response reduction are presented in this paper. A structural system with semi-active control devices under random loading is modelled as a controlled, randomly excited and dissipated Hamiltonian system of multi-degree of freedom. The control force produced by a semi-active control device is split into semi-active part and passive part incorporated in the uncontrolled system. Applying the statistical linearization method to the non-linear multi-degree-of-freedom system with passive control force components yields quasi-linear equations of motion, which can tend to corresponding linear ones with system response reduction. By applying the dynamical programming principle to the controlled linearized system, a dynamical programming equation is established and in particular, for a non-filtering white noise excitation, is solved as an optimal regulation problem to determine the quasi-linear quadratic optimal control law and furthermore semi-active optimal control law according to the variational principle. Then the semi-active optimal control of a tall building structure with magnetorheological-tuned liquid column damper (MR-TLCD) under random wind excitation is performed by using the proposed method. The non-linear model of the structural system with semi-active MR-TLCD is formulated in structural mode space and uncoupled between structural and MR fluid accelerations. The quasi-linear equations for system states are derived from the model and the dynamical programming equation for the system is obtained. In the case that the random wind excitation with the Davenport power spectrum cannot be modelled as a linear filtering white noise, the dynamical programming equation is solved as an optimal regulation problem to obtain the semi-active optimal control force, on which the clipping treatment may be performed to ensure the control force implementable actually. Eventually, the response statistics of the semi-actively controlled structure under random wind excitation are evaluated by using the statistical linearization method, and are compared with those of the passively controlled structure to determine the control efficacy. Numerical results illustrate the high control effectiveness of the proposed semi-active optimal control method for building structures with MR-TLCDs.