This note presents simple controllers for achieving asymptotic trajectory (set-point and time-varying) tracking under the conditions of restricted input in vertical takeoff and landing aircraft. Unlike previous studies that apply nonlinear combination of saturated functions for stabilizing the closed-loop system, the proposed controllers are based on smooth functions that can easily be implemented in real time applications.Index Terms-Bounded input, global stability, semiglobal stability, trajectory tracking, vertical take-off and landing (VTOL) aircraft.

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In a recent paper Tomei presented a globally asymptotically stable PD regulator for robots with flexible joints. A drawback of this scheme is that noise in velocity measurements degrades performance. In this paper we show that velocity measurement in Tomei's scheme can be replaced by approximate differentiation preserving global asymptotic stability. Simulations that illustrate our result are also presented'. 2 Main Result Propositioii 1. Consider the n degree of freedom flexible joints robot model (1) in closed-loop with the following controller: where ai and bi are positive constants, and l i p , K D are constant diagonal positive definite matrices. Assume I\ ' and K p satisfy min{A,i,(K), &,,in(#p)} > p6. Under these conditions, the closed loop system has * T T . T , $ t ' T = an unique equilibrium point: [qT, q1 , q2 , q2 [qTd,O, qTd, 0, OIT which is GAS. ] 1 U.S. Government work not protected by U.S. copyright 617

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