# Comparative Study Between Active AMD and ABS Devices by Using \(\mu\)-Synthesis Robust Control
###### Abstract
The controller design process within a robust control system follows a structured method to create a robust controller capable of maintaining performance even under extreme conditions or variations, making it particularly useful in applications such as seismic control or other high-stakes engineering systems. These steps include the following:  
* Identifying the system uncertainties by modeling the uncertainties present in the system;
* Setting performance and stability specifications, which involves defining performance objectives, such as minimizing vibration amplitude and establishing stability margins that the controller must achieve despite uncertainties;
* Using \(\upmu\)-analysis for robustness assessment with the modeled uncertainties;
* Applying \(\upmu\)-analysis to measure the system's robustness;
* Synthesizing a controller using iterative synthesis, which involves minimizing the worst-case value across all the frequencies. The structured singular value \(\upmu\) indicates how much uncertainty the system can withstand before performance degrades or instability occurs. This helps determine the level of robustness the controller needs to achieve. The primary aim of our robust control system for structural applications is to minimize displacement under external disturbances, such as seismic events. We achieved this by setting performance objectives that limit the allowable displacement and optimizing the controller to maintain movement within these limits, thereby ensuring stability and safety. We also imposed constraints on the control force to prevent actuator saturation and reduce energy consumption. Through careful tuning of the controller gains and the use of a cost function that penalizes excessive control force, we achieved a balance between robustness and performance. Our method successfully integrated these goals, achieving effective damping while minimizing both structural displacement and control force, resulting in a robust response to the disturbances. Some researchers have proposed a simple control force estimation approach to reduce control costs. The authors in [21] developed a response spectrum-based design strategy to incorporate active control, base isolation, and damping technologies. Viscous dampers dissipate energy through fluid motion, while hysteretic dampers absorb energy through material deformation. Similarly, [22] presents a spectral technique that relates the structural response of base-isolated buildings to seismic inputs, simplifying the estimation of LQR control force for passive base-isolated structures. This paper examines the application of \(\upmu\)-synthesis, a robust control law, to actively regulate the response of three-story buildings to actual earthquakes, utilizing AMD and ABS as the primary active systems. Consequently, this study seeks to address this issue by comparing the two primary active control methods, ABS and AMD. We utilize the renowned efficiency of robust synthesis control, effectively implemented in several engineering applications over the past decade [23; 24; 25], to mitigate seismic impacts on structures. Indeed, the selected control is renowned for its ability to incorporate various practical considerations simultaneously in the form of achievable performance objectives. The method's ability to handle and measure uncertainties from various sources, as well as translate performance goals into clearly stated weighting functions in the control law criteria, makes it highly appealing [1; 23]. So, we performed a comparison study on how to control the seismic vibration response of a building model that includes both parametric and dynamic uncertainties. Our main goals were to find the best way to reduce the building's displacement and control the force that would be needed. In this study, we aimed to reduce the response of a three-degree-of-freedom uncertain model of a building subjected to the earthquake of El Centro (1979) via the application of \(\mu\)-synthesis control.