Optimal adaptive nonholonomic stabilisation

One of Zhong-Ping Jiang's early-career seminal results have been on the trajectory tracking feedback design for chained systems, and for the nonholonomic unicycle in particular. However, for equilibrium stabilisation (parking), globally stabilising feedback laws are exceedingly rare. Just as rare are closed-form optimal controllers for nonlinear systems, which avoid the need to solve – and store the solutions of – Hamilton-Jacobi-Bellman PDEs. In fact, until our very recent work, for the unicycle stabilisation problem, closed-form optimal controllers, even of the inverse optimal kind, have been nonexistent. Another subject in which Zhong-Ping has made pioneering contributions, over the last decade and a half, is adaptive optimal control, also referred to by the moniker reinforcement learning. An interest from my early career has commuted adjectives – optimal adaptive control, where optimality holds over the entire infinite time horizon, rather than just as the time approaches infinity. In this short paper I celebrate Zhong-Ping with a result that pulls together his various interests in nonholonomic, adaptive, and optimal control with my interests in these subjects, spanning three decades. For unicycles with large parametric uncertainties on both inputs, I present designs that are both adaptive and infinite-horizon optimal, while also being globally stabilising. This is achieved by constructing strict control Lyapunov functions, using backstepping, forwarding, or passivity, and then taking adaptive LgV controllers, which are inverse optimal.