Nonlinear Tracking Control of the VTOL AircraftAbstract :In this paper, we present a nonlinear tracking controller for the nonminimum phase, underactuated model of a vertical take off and landing (VTOL) aircraft. Specifically, the controller is designed to ensure that the VTOL aircraft position/orientation tracks a reference signal generator. The controller ensures that the position/orientation tracking error can be exponentially forced into an arbitrarily small neighborhood around zero (i.e., Globally Uniformly Ultimately Bounded (GUUB) tracking). Past recearch :Over the last decade, there has been considerable interest with regard to designing controllers for the VTOL aircraft. The VTOL control problem is interesting because the simplified two degree of freedom dynamics are nonlinear, nonminimum phase, and underactuated. It has been acknowledged that static inputoutput linearization approaches to decouple rolling moment and the lateral acceleration seem to fail to produce satisfactory performance as the type of approach often results in the unstable roll dynamics being unobservable (The reader is referred to [7] for more background on this problem). To attack this problem, Hauser et al. in [7] proposed an approximate inputoutput linearization; however, the controller was initially designed by assuming that the coupling between the thrust and the rolling moments can be neglected. Unfortunately, this assumption leads to magnitude restrictions on the coupling constant; furthermore, the approach also requires that the roll of the VTOL aircraft be restricted to the interval (90^{o},+90^{o}). In [9], Martin et al., proposed a tracking controller based on differential flatness. The author used an interesting approach by noting that the output at a point fixed with respect to the aircraft body (Huygens center of oscillation) can be used, which results in a flat inputstate system. This result did not impose any restrictions on the coupling coefficients nor on the roll of the VTOL aircraft; however, this controller is not defined over the entire state space. In [8], Lin et al. designed a setpoint controller using optimal control techniques; however, the control law required similar restrictions as those in given [7] (in addition, the control design methodology assumed that a part of the control input can be bounded as a state disturbance). In [10], McClamroch et al. proposed hybrid switching strategies for two different setpoint control problems. In [1], dynamic inversion and robust control techniques were used to deal with the nonminimum phase dynamics; however, this type of approach imposed restrictions on the desired trajectory. In [2], output tracking and maneuver regulating controllers were proposed to illustrate the advantages of a maneuver regulating controller over a output tracking approach. In [3], nontrivial extensions of [1] and [2] were presented for the CTOL (conventional take off and landing) problem. In [11] and [12], Oishi et al. proposed a fusion of approximate linearization and switching techniques to develop tracking controllers in a ''safe'' envelope; however, some of the aforementioned restrictions were not completely avoided. Recently, SiraRamirez et al., [13] proposed an approximate solution to the general reference trajectory tracking problem. In this paper, we use some our previous work in underactuated systems given in [4] and [6] to design a nonlinear state tracking controller for the VTOL problem that ensures Globally Uniformly Ultimately Bounded (GUUB) tracking (i.e., state tracking with respect to a reference signal generator). The approach uses a series of transformations similar to the ones provided in [9] and [4] to manipulate the VTOL dynamics into a suitable form which allows a Lyapunovbased controller to be designed. The controller uses a dynamic oscillator embedded inside of the overall control strategy to provide additional design flexibility. Simulation ResultsThe proposed controller was simulated for a VTOL aircraft which gave us the following results Position tracking error for the Xcoordinate Position tracking error for the Ycoordinate Control torque input: a) steering motor and b) drive motor
