# Earthquake Hazard Mitigation for Uncertain Building Systems Based on Adaptive Synergetic Control
###### Abstract
This control concept, based on synergetic theory, permits the trajectories of the controlled system to evolve on an invariant manifold selected by the designer and to obtain the desired performance in the simultaneous presence of parametric and nonparametric uncertainties. The following general procedure can be followed in designing nonlinear systems based on synergetic control [23; 24; 25; 26; 27; 28]:  
* Forming an extended differential equation system that represents the various operations, such as optimizing, achieving fixed values, suppressing disturbances, observing coordinates, etc. ;
* Synthesizing _external_ controls that work to reduce the extra degree of freedom (DOF) of the extended dynamic system model relative to the final manifold;
* Synthesizing the _internal_ controls, which can establish links or relations among the _internal_ coordinates of the system. The control goal can be guaranteed and assured by these links. In view of synergetic theory, the synergetic controller guides the trajectories of dynamic system trajectories to move, onto the manifold, from any initial states to their equilibrium states. In this study, synergetic control theory was applied to develop a novel nonlinear control approach to mitigate the vibration in an MR-damper-supported building structure due to earthquake. Two SC-based controllers were designed to solve the control problem in earthquakes buildings. The conventional synergetic controller (CSC) is the first candidate controller, while the second controller was established on the basis of the adaptive synergetic controller (ASC). In adaptive control techniques, the parameters of a plant in real time are adjusted in order to maintain a desired level of dynamic performance when the system is subjected to unknown and varying (changing with time) parameters [29; 30; 31; 32; 33]. The adaptive version of control based on the synergetic methodology is devoted to solving the problem of uncertainties in the construction device's parameters due to the effect of earthquake by actuating the necessary force via the MR damper. The stability of the earthquakes controlled structure was analyzed and proven on the basis of Lyapunov theory. The contributions of the work can be summarized by the following points:  
* Developing classical and adaptive synergetic control algorithms to solve the vibration control problem in earthquakes buildings on the basis of MR dampers and Lyapunov stability analysis;
* Proving asymptotic stability for building systems controlled by classical and adaptive synergetic control schemes, such that all errors finally converge to their corresponding zero equilibrium points based on the Lyapunov theorem;* Guaranteeing the boundedness of the estimated viscosity and stiffness coefficients of building systems;
* Conducting a comparison between the proposed classical and adaptive synergetic controllers in terms of vibration suppression capabilities. The rest of the article is organized as follows: A dynamic model of a one-story building is developed in Section 2, and the control methodology based on synergetic control to suppress the vibrating action of an earthquake is analyzed for both classical and adaptive schemes in Section 3. In Section 4, the results of a computer simulation conducted to verify the effectiveness of the proposed control scheme are presented. The concluded points based on observations of the simulated results are highlighted in Section 5. This section also includes suggestions for future extension.