Abstract
In this paper, we evaluate theoretical aspects of a distributed system of noncooperative robots controlled by a distributed model predictive control scheme, which operates in a shared space. Here, for collision avoidance, the future predicted state trajectories are projected on a grid and exchanged via discrete cell indexes to reduce the communication burden. The predicted trajectories are obtained locally by each robot and carried out in the continuous space. Therefore, the quantisation does not impose the quality of the solution. We derive sufficient conditions to show convergence and practical stability for the distributed control system by using an idea of a temporary roundabout derived from crossing patterns of street traffic rules, which is established in a fixed and flexible circle size. Furthermore, a condition for the sufficient prediction horizon length to recognise necessary detours is presented, which is adapted for the occupancy grid. The theoretical results match with the trajectory patterns from former numerical simulations, showing that this pattern is naturally chosen as an overall solution.
Highlights
Today, control problems with high dynamics have been of high interest in production and logistic processes or in traffic scenarios
Prediction horizon length with an occupancy grid In Model Predictive Control (MPC), an appropriate prediction horizon length is crucial for convergence
If the distance between two robots is such that Equation (28) is fulfilled and Assumption 5 is satisfied for all robots p ∈ [1 : P], the calculation of a circle and execution of the Distributed Model Predictive Control (DMPC) scheme in Algorithm 2 using the switched control by Algorithm 4 will allow the robots to let them converge to their targets, i.e., to steer the system practically stable
Summary
Control problems with high dynamics have been of high interest in production and logistic processes or in traffic scenarios. The authors in Reference [28] applied a Lyapunov controller with a global bounded state feedback controller for a tracking problem of a non-holonomic robot In these articles, the convergence or stability aspects were shown by utilising either a Lyapunov function or a controller, which guarantees the decrease of costs in any time instant, or of terminal constraints or costs, which forces the predicted trajectory to achieve an end point or region, or to steer that by a terminal cost penalty or a connective constraint, ensuring that, e.g., a minimum distance or explicit bounds on the optimal value function were used to calculate the suboptimality degree of a finite horizon controller. For a vector x ∈ Rn , n ∈ N, we define the infinity norm k x k∞ := maxi∈[1:n] | xi |
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