Semiglobal exponential stability of a counter-current and co-current guidance scheme

A control technique for counter-current and co-current guidance of underactuated marine vehicles is revisited and stronger stability properties are shown. In particular, the stronger property of uniform semiglobal exponential stability is shown for the complete multiple-equilibria closed loop system, that has previously been shown to be uniformly semiglobally asymptotically stable and uniformly locally exponentially stable. Compared to the original proof, the analysis presented in this paper does not invoke the theory developed for cascaded systems; it follows instead a direct approach where a Lyapunov function for the full system is identified. This shows that analysing stability of a complex non-linear system by means of proper Lyapunov function candidates can yield significant results, leaving however the designer with the challenging task of identifying the right candidate. The theory is supported by simulations.