State estimation in mechanical systems of fractional-order based on a family of Proportional $\rho$-Integral observers

In control theory, there are many proposals to solve the problem of observer design. This paper studies the Mittag-Leffler stability of a class of dynamic observers for nonlinear fractional-order systems, given in the observable canonical form and defined by the Caputo fractional derivative. We prove that the Riemann-Liouville integral could be employed to provide robustness against noisy measurements during the estimation problem. Based on this advantage, the main result of this paper consists of the design of a family of high gain proportional rho-integral observers employed to estimate unmeasured state variables of nonlinear fractional systems of commensurate order. Three illustrative numerical examples of mechanical systems are provided, which corroborate the effectiveness of the proposed algorithms.