Stability of Actively Controlled Structures with Actuator Saturation

Because of the stochastic nature of earthquakes and limited capacities of actuators, it is conceivable that the actuators used for the control of civil engineering structures may be saturated during strong earthquakes. In this paper, the stability of actively controlled linear structures under a variety of popular control methods during actuator saturation is investigated. Sufficient conditions to guarantee the asymptotic stability of the structure during actuator saturation are examined and discussed. These conditions involve the solution of a system of simultaneous linear matrix inequalities (LMI). A simple and efficient computer code to solve a system of LMI, based on the MATLAB LMI toolbox, is presented for use by the readers. Based on sufficient conditions, a method for designing general controllers, that guarantee asymptotic stability during actuator saturation, is presented. It is shown analytically that Lyapunov controllers, H∞-type controllers, and sliding-mode controllers are asymptotically stable during actuator saturation. For multi–degrees-of-freedom (MDOF) systems using pole assignment and linear quadratic regulator (LQR) controllers, it has been shown through extensive simulations results that: (1) asymptotic stability of structures is more likely to be guaranteed for low-gain controllers than for high-gain controllers; (2) adding active damping to the structure is much more beneficial than adding active stiffness, in terms of asymptotic stability; and (3) asymptotic stability can be guaranteed for quite high levels of added active damping to various modes of the structure. Further extensive simulation results demonstrate that static output feedback controllers using only velocity feedback are always asymptotically stable in the range of practical applications. Likewise, for static output controllers using collocated sensors, the region of asymptotic stability is quite large.