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The development of a novel adaptive autopilot for the dive-plane control of multi-input multi-output submarines with unmodeled dynamics, based on the ℒ1 adaptive control theory, is the subject of this article. An ℒ1 adaptive autopilot is designed for the trajectory control of the depth and pitch angle using bow and stern hydroplanes. Interestingly, the structure of the adaptive controller remains fixed, regardless of the nonlinearities and external disturbance inputs, retained in the model of the submarine. Unlike the traditional adaptive control laws, the ℒ1 adaptive control input is generated by filtering the estimated control signal. A nice feature of the control law is that it is possible to achieve fast adaptation and desirable performance bounds in the closed-loop system by the choice of large adaptation gains. Simulation results are presented, which show that the autopilot accomplishes precise trajectory control in the dive plane, despite parametric uncertainties, unmodeled nonlinearities, and random disturbance inputs.

This paper treats the question of adaptive control of prototypical aeroelastic wing sections with structural nonlinearity based on the immersion and invariance approach. The chosen dynamic model describes the nonlinear plunge and pitch motion of a wing. A single control surface is used for the purpose of control. It is assumed that the model parameters except the sign of coefficient of control input are unknown. A noncertaintyequivalent adaptive control law for the trajectory tracking of the pitch angle is derived. Using Lyapunov analysis, asymptotic convergence of the state variables to the origin is established. Unlike the certainty-equivalent control laws developed in literature for aeroelastic systems, this new control system can accomplish superior tracking performance. A special feature of the designed control system is that, whenever the estimated parameters coincide with their true values, the adaptation stops and the closed-loop system recovers the performance of deterministic closed-loop system. This cannot happen if certainty-equivalent adaptive controllers are used. Furthermore, the trajectory of the closed-loop system, including the noncertainty-equivalent adaptive law, is eventually confined to a manifold in the space of state variables and parameter estimates. Simulation results using the new controller and the conventional certainty-equivalent controller are presented. These results show that the new controller performs better in suppressing oscillatory responses of the wing in the presence of large parameter uncertainties. Nomenclature a = nondimensionalized distance from the midchord to the elastic axis b = semichord of the wing c h = structural damping coefficient in plunge due to viscous damping c = structural damping coefficient in pitch due to viscous damping c 1 , = control gain and weighting matrix h = plunge displacement I = mass moment of inertia of the wing about the elastic axis k h = structural spring constant in plunge k = structural spring constant in pitch M, g 0 , b 0 = system matrices m t = mass of the plunge-pitch system m w = mass of the wing s p = span U = freestream velocity V i , V = Lyapunov functions x = states x = nondimensionalized distance measured from the elastic axis to the center of mass z i = parameter errors = pitch angle r = pitch angle reference trajectory = flap deflection i = parameter used in update laŵ i = partial parameter estimates i , = nonlinear function used in parameter error expression = density of air

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