Modeling and Optimal Control of the Vibration Model of Constrained Buildings Based on Fractional Order Singular Theory Using Orthogonal Polynomials

This paper introduces the optimal control of fractional-order singular systems and the application of finite-mode systems in modeling practical systems. In this paper, there are only finite modes in the system. The aim is to control fractional order singular systems without using operational matrices using optimal control and virtual calculus of variations. After using orthogonal polynomials, we achieved a formulation to find the answer to the problem. This problem was solved using Legendre, Chebyshev, and Jacobi orthogonal polynomials. Then, the vibration model of two constrained buildings is modeled using the fractional order singular system. The aim is to control the vibration of the constrained buildings against earthquakes. Finally, a comparison was made between different orthogonal polynomials and the number of the polynomial of these functions.