Longitudinal Stability Analysis of a UAV under the Uncertainty of Two Stability Derivatives

The longitudinal stability analysis of an aircraft is performed by the investigation of root locations of its transfer function’s denominator (the characteristic equation). However, this transfer function is obtained by linearizing aircraft dynamic model at a certain operation point (altitude and speed). However, aircraft have varying stability derivatives, therefore dynamic behavior, for different flight phases such as take-off, cruise, and landing. Thus, the stability investigation of the characteristic equation can be said to be valid only for a certain flight condition. In reality, stability derivatives have varying values depending on flight conditions. Therefore, an analysis including all possible values of stability derivatives in the flight envelope is required to guarantee stability. In this study, two most varying stability derivatives in the transfer function were taken as uncertain parameters. Gridding these two parameters to check the stability of the UAV for all possible flight conditions can be thought as a method, but it is very time-consuming, and it cannot assure the stability theoretically. A new simple approach, guaranteeing stability under the uncertainty of two stability derivatives, is developed by using the Edge and Bialas theorems. Here, the problem of the investigation of the stability under the uncertainty of two stability derivatives is reduced to the analysis of four polynomials. Thus, the stability characteristics of an airplane for a given flight envelope can be easily determined by just looking at the eigenvalues of the matrices obtained from these four polynomials.