This paper deals with the formation control problem without collisions for second-order multi-agent systems. We propose a control strategy which consists of a bounded attractive component to ensure convergence to a specific geometrical pattern and a complementary repulsive component to guarantee collision-free rearrangement. For convergence purposes, it is assumed that the communication graph contains at least a directed spanning tree. The avoidance complementary component is formed by applying repulsive vector fields with unstable focus structure. Using the well-known input-to-state stability property a control law for second-order agents is derived in a constructive manner starting from the first-order case. We consider that every agent is able to detect the presence of any other agent in the surrounding area and also can measure and share both position and velocity with his predefined set of neighbours. The resulting control law ensures the convergence to the desired geometrical pattern without collisions during the transient behaviour, as well as bounded velocities and accelerations. Numerical simulations are provided to show the performance and effectiveness of the proposed strategy.

In this paper, we investigate the interval observer problem for a class of discrete-time nonlinear systems, in absence or presence of external disturbances and parametric uncertainties. The interval observers depend on the design of two preserving order observers, providing lower and upper estimations of the state. The main objective is to apply the stability radii notions and cooperativity property in the estimation error systems in order to guarantee that the lower/upper estimation is always below/above the real state trajectory at each time instant from an appropriate initialization, and the estimation errors converge asymptotically towards zero when the disturbances and/or uncertainties are vanishing. For the disturbed case, the estimation errors practically converge to a vicinity of zero, while the lower/upper estimations preserve the partial ordering with respect to the state trajectory. The design conditions, that are valid for Lipschitz nonlinearities, can be expressed as Linear Matrix Inequalities (LMIs). A numerical simulation example is provided to verify the effectiveness of the proposed method.

This paper presents a new approach to design preserving order and interval observers for a family of nonlinear systems in absence and in presence of parametric uncertainties and exogenous disturbances. A preserving order observer provides an upper/lower estimation that is always above/below the state trajectory, depending on the partial ordering of the initial conditions, and asymptotically converges to its true values in the nominal case. An interval observer is then constituted by means of an upper and a lower preserving order observer. In the uncertain/disturbed case, the estimations preserve the partial ordering with respect to the state trajectory, and practically converge to the true values, despite of the uncertainties/perturbations. The design approach relies on the cooperativity property and the stability radii mathematical tools, both applied to the estimation error systems. The objective is to exploit the stability radii analysis for the family of linear positive systems under the time-varying nonlinear perturbations in order to guarantee the exponential convergence property of the observers, while the cooperativity condition determines the partial ordering between the trajectories of the state and the estimations. The proposed approach, defined for Lipschitz nonlinearities, depends only on two observer matrix gains. The design is reduced to the solution of linear matrix inequalities, which are given by the cooperative condition and convergence constraints. An illustrative example is presented to show the effectiveness of the theoretical results.