Stability of switched linear differential systems

We study the stability of switched systems where the dynamic modes are described by systems of higher-order linear differential equations not necessarily sharing the same state space. Concatenability of trajectories at the switching instants is specified by gluing conditions, i.e. algebraic conditions on the trajectories and their derivatives at the switching instant. We provide sufficient conditions for stability based on LMIs for systems with general gluing conditions. We also analyse the role of positive-realness in providing sufficient polynomial-algebraic conditions for stability of two-modes switched systems with special gluing conditions

Optimisation of Low-Thrust and Hybrid Earth-Moon Transfers

This paper presents an optimization procedure to generate fast and low-∆v Earth-Moon transfer trajectories, by exploiting the multi-body dynamics of the Sun-Earth-Moon system. Ideal (first-guess) trajectories are generated at first, using two coupled planar circular restricted three-body problems, one representing the Earth-Moon system, and one representing the Sun-Earth. The trajectories consist of a first ballistic arc in the Sun-Earth system, and a second ballistic arc in the Earth-Moon system. The two are connected at a patching point at one end (with an instantaneous ∆v), and they are bounded at Earth and Moon respectively at the other end. Families of these trajectories are found by means of an evolutionary optimization method. Subsequently, they are used as first-guess for solving an optimal control problem, in which the full three-dimensional 4-body problem is introduced and the patching point is set free. The objective of the optimisation is to reduce the total ∆v, and the time of flight, together with introducing the constraints on the transfer boundary conditions and of the considered propulsion technology. Sets of different optimal trajectories are presented, which represents trade-off options between ∆v and time of flight. These optimal transfers include conventional solar-electric low-thrust and hybrid chemical/solar-electric high/low-thrust, envisaging future spacecraft that can carry both systems. A final comparison is made between the optimal transfers found and only chemical high-thrust optimal solutions retrieved from literature

On Robust Stability of Limit Cycles for Hybrid Systems with Multiple Jumps

In this paper, we address stability and robustness properties of hybrid limit cycles for a class of hybrid systems with multiple jumps in one period. The main results entail equivalent characterizations of stability of hybrid limit cycles for hybrid systems. The hybrid limit cycles may have multiple jumps in one period and the jumps are allowed to occur on sets. Conditions guaranteeing robustness of hybrid limit cycles are also presented

Stably Extending Two-Dimensional Bipedal Walking to Three Dimensions

In this paper we develop a feedback control law that results in stable walking gaits on flat ground for a three-dimensional bipedal robotic walker given stable walking gaits for a two-dimensional bipedal robotic walker. This is achieved by combining disparate techniques that have been employed in the bipedal robotic community: controlled symmetries, geometric reduction and hybrid zero dynamics. Controlled symmetries are utilized to obtain stable walking gaits for a two-dimensional bipedal robot walking on flat ground. These are related to walking gaits for a three-dimensional (hipless) bipedal robot through the use of geometric reduction. Finally, these walking gaits in three dimensions are made stable through the use of hybrid zero dynamics