Null controllability and stabilization of linear systems subject to asymmetric actuator saturation

This paper generalizes our recent results on the null controllable regions and the stabilizability of exponentially unstable linear systems subject to symmetric actuator saturation. The description of the null controllable region carries smoothly from the symmetric case to the asymmetric case. As to stabilization, we have to take a quite different approach since the earlier development relies mainly on the symmetric property of the vector field. Specifically, in this paper, we construct a Lyapunov function from a closed trajectory to show that this closed trajectory forms the boundary of the domain of attraction for a planar anti-stable system under the control of a saturated linear feedback. If the linear feedback is designed by the LQR method, then there is a unique limit cycle which forms the boundary of the domain of attraction. We further show that if the gain is increased along the direction of the LQR feedback, then the domain of attraction can be made arbitrarily close to the null controllable region. This design can be utilized to construct state feedback laws for higher order systems with two exponentially unstable poles