## 1 Introduction  
Many researchers [1; 2; 3] have investigated smart structures and the suppression of oscillations caused by external disturbances. The field of smart structures has significantly developed in recent years [4; 5; 6; 7]. Damping oscillations require the use of smart materials and the application of controls [8; 9; 10; 11; 12]. Advanced verification techniques must be used to account for modeling uncertainties and imperfections. Such techniques can include \(\mu\)-analysis, \(\mu\)-synthesis, and robust control. In this study, we used the above types of controls and applied them to smart piezoelectric structures. H-infinity feedback optimal output (Hifoo) control in smart piezoelectric structures involves designing controllers that minimize the impact of disturbances and uncertainties on the output performance of the system. This is particularly important in applications where precise control of the deformation or vibration of the structure is required. Modeling with Hifoo assumes infinite control knowledge. Hinfinite control is a common approach for vibration control [13; 14; 15; 16]. Hifoo is related to H-infinity control, which aims to minimize the effects of disturbances and uncertainties in a system. Hifoo is specifically focused on finding optimal controllers to achieve the best possible output performance. The key benefits of applying \(\mu\)-analysis control to smart piezoelectric structures include improved robustness against external disturbances, increased stability, and enhanced tracking of the desired responses. These controllers are designed to optimize the trade-off between performance and robustness, making them suitable for applications in which maintaining a specific structural response is critical. A combination of \(\mu\)-analysis and Hifoo [4; 5; 6; 17] reduced control theory was utilized to study the design of piezoelectric active control for both normal and damaged smart buildings. The outcomes validated the efficacy of the proposed model and proceduresand the control behavior of the beam aligned with expectations. Following a thorough study, we assessed the robustness and performance of the system. By adding uncertainty, we were able to maintain the framework within specified uncertainty bounds [7; 18; 19; 20]. In a research paper that explored the benefits of robust control in smart structures, refs. [19; 20; 21; 22; 23; 24] undersored the utilization of Hifoo control [4; 5; 6; 17] in both the state space and frequency domains. The advantages of this work are as follows: the introduction of control for oscillation suppression and the modeling of intelligent structures bring about alterations in both the frequency and time-space domains, as well as the inclusion of uncertainties in the mathematical model of the construction, and an introduction to \(\mu\)-synthesis and reduced-order control in intelligent structures [25; 26]. The development of control strategies for intelligent piezoelectric structures involves several challenging issues. Researchers have investigated the application of piezoelectric materials in systems with distributed parameters with the aim of facilitating efficient and cost-effective active control. Dynamic systems can be actively controlled using distributed sensors and actuators composed of adaptable piezoelectric materials. The main considerations that structural control engineers need to keep in mind while creating reliable control techniques for assessing resilience, optimal placement, and structural modeling in the face of uncertainty are covered in this essay [27; 28; 29; 30]. Because the controller provided is of order 56, sophisticated control techniques may be applied to simpler models, and we use the optimization method Hifoo to reduce the order of the controller. All simulations were completed using sophisticated programming methods in the Matlab software platform (v. 8.0, Mathworks, Natick, MA, USA). A crucial engineering problem is the vibration suppression that occurs under dynamic and unexpected loads.