Adaptive control design for underactuated systems using sums-of-squares optimization

Few methods have been proposed for designing adaptive controllers for underactuated systems with constant, unknown parameters. This is primarily because nonlinear underactuated systems are typically not feedback linearizable; most model-reference adaptive control approaches for nonlinear systems, especially those involving Lyapunov analysis, rely heavily on feedback linearization to guarantee stability. Self-tuning regulators do not have this limitation, but these more flexible methods are often computationally intractable and usually lack a proof of stability. Here, we propose an alternative adaptive control design procedure which can handle underactuated systems of moderate dimension (≤ 12). By making use of recent advances in sums-of-squares optimization, we build upon the work done in [31] and [10], to design adaptive controllers with verified robustness to parameter uncertainty, and time-varying adaptive controllers with guaranteed finite-time performance along system trajectories. We demonstrate our algorithm on three simulated underactuated systems - the Acrobot and cart-pole systems with unknown viscous friction terms and the perching glider with unknown aerodynamic coefficients.