Global stabilization of a PVTOL aircraft model with bounded inputs

Abstract

Abstract We propose a global stabilizing control design for the planar vertical takeoff and landing (PVTOL) aircraft, with bounded inputs. The approach is based on the use of non-linear combinations of linear saturation functions bounding the thrust input and the rolling moment to arbitrary saturation limits. We provide global convergence of the state to the origin, using a relatively simple algorithm. Notes † In deriving (Equation1), every force acting on the PVTOL is considered to be normalized by the aircraft's weight (e.g. , where F, m and g respectively represent the actual thrust acting on the PVTOL, the aircraft's mass and the gravity acceleration), while torques are dimensional according to their (angular) acceleration effect (in relation to the aircraft's moment of inertia J, i.e. , where τ represents the actual rolling torque acting on the PVTOL). See, for instance Fliess et al. (Citation1999), where the derivation of (Equation1) is developed and the definitions of are given in terms of the system parameters and variables. Note that, under such considerations, and the normalized gravity acceleration are dimensionless, while u 2 is expressed in angular acceleration units. Further, x, y, and are not properly expressed in position and velocity units, but these are expressed in relation to the value of the gravity acceleration (e.g. , where x c and y c respectively represent the actual horizontal and vertical positions of the aircraft's centre of mass). ‡ For example, ϵ = 0.01 is typical during jet-borne flight, i.e. hover, for the YAV-SB Harrier (produced by McDonnell Aircraft Company see Hauser et al. (Citation1992, p. 671): a description of the Harrier can be found in Hauser et al. (Citation1992, §2.1)); see references therein for more detailed information about such a V/STOL (vertical/short takeoff and landing) aircraft. † Though, strangely enough, and are neglected in the control law when a stabilization objective equivalent to the one adopted here is considered. ‡ Hence, such a property () turns out to be redundant in Definitions 1 and 2 of Teel (Citation1996). † A 2-level linear saturation for () is a simple linear saturation (Teel Citation1992, Definition 2) if and only if and . † Such an assumption will be proved to be satisfied with in the second part of the proof. † Recall that this was assumed in the first part of the proof. It is then demonstrated that such an assumption is actually a fact, confirming the validity of (Equation24) and (Equation25). † Note however that since E 1 and E 2 were obtained by considering extreme values of each term in (Equation19) and (Equation20), such criterion is expected to furnish extremely small values of k which lead to extremely slow closed-loop system performances, specially concerning the horizontal motion of the PVTOL aircraft. A less restrictive criterion for k can still be obtained through the expressions and relations developed previously.