Globally stabilizing quasi time-optimal control of planar saturated linear systems

We propose a family of nonlinear state feedback global stabilizers for all planar linear systems which are globally stabilizable by bounded inputs (namely, all non exponentially unstable linear systems). This family is parametrized by a nonlinear function whose selection has great impact on the closed-loop performance. Special selections correspond to linear global stabilizers, which are well known lead to poor performance, especially when the underlying system is not exponentially stable, or weakly exponentially stable. We show that in some relevant cases (e.g., the double integrator case) suitable selections of this gain correspond to quasi time-optimal responses, where the ldquoquasirdquo is required to guarantee local exponential stability of the closed-loop. Simulation results show that the arising trajectories are quasi time-optimal for arbitrarily large initial conditions which, in light of the very simple nonlinear control law, are very useful for embedded control application with strong computational constraints.