Adaptive Dynamic Surface Control for Unmanned Aerial Vehicles Based on Attractive Manifolds

Abstract

T RADITIONALLY, linear flight controllers are designed based on multiple trimmed flight conditions and combined using gain scheduling to guarantee the performance in the entire flight envelope [1]. However, this process requires extensive offline analysis and flight testing. Advanced nonlinear control methods are proposed to overcome the shortcoming of linear design approaches, such as feedback linearization, backstepping, and dynamic surface control. The feedback linearization approach [2] requires an accurate aircraft mathematical model. However, it is rather difficult to exactly model the aerodynamic characteristics. Therefore, the adaptive feedback linearization method [3] with an adaptive outer loop to compensate for uncertainties in the aircraft model has been developed. The problem of this method is that it does not give any closed-loop stability or tracking-error convergence guarantees [1]. One nonlinear adaptive control method that does give these guarantees is the adaptive backstepping design approach. The adaptive backstepping method [4,5] makes use of the step-bystep coordinate transformation and uses some states as intermediate virtual controls to regulate other states. This method relies on finding an adaptive control law and a parameter update law to yield that the Lyapunov function of the closed-loop system is negative. However, the dynamics of the estimation error can not be prescribed directly, which may cause the undesired transient response of the closed-loop system [6]. Recently, a new method for stabilization and adaptive control of uncertain nonlinear systems based on immersion and invariance methodology has been proposed [7–9]. This newmethod relies upon the notion of attractive manifolds on which the dynamics of the closed-loop system is independent of unknown parameters. This method allows for prescribing the dynamics of the parameterestimation error, and the parameter-estimation process automatically stops if the parameter estimate happens to converge to its corresponding true value. However, this approach relies on the solution of the partial differential equation, which is difficult for multivariable systems. To overcome the problem of solving the partial differential equation, amodification of the immersion and invariancemethod has been developed [10,11]. This extended immersion and invariance method also relies on the notion of attractive manifolds and introduces a stable linear filter for the regressor matrix to avoid solving the partial differential equation. This adaptive control law based on attractive manifolds is combined with the backstepping method and applied to the simplified F-18 model [12]. However, this method suffers from the problem of “explosion of complexity” arising from the differentiation of the intermediate virtual control. To overcome this problem, the dynamic surface control has been proposed [13,14]. This approach is similar to the backstepping method, but introduces a low-pass filter to prevent calculation of the analytic derivative of virtual control functions at each design step. In this work, an adaptive flight-control law based on attractive manifolds has been derived for unmanned aerial vehicles (UAVs). The controller employs the dynamic surface control to avoid “explosion of complexity.”The extended parameter estimation based on attractive manifolds is used to sidestep the solving of the partial differential equation. This adaptive flight-control law is applied to the fully six-degree-of-freedom nonlinear UAV model with some uncertain aerodynamic parameters.