This paper describes hybrid model predictive controllers that switch between two predictor functions based on the uncontrollable divergence metric. The uncontrollable divergence metric relates the computational capabilities of the model predictive controller, to the error of the system due to model mismatch of the predictor function during computation of the solution. The contribution of this paper is in its treatment of the model predictive controller to permit optimization to take multiple timesteps to occur, but still rely on the uncontrollable divergence metric. The results demonstrate the approach for control of a vertical takeoff-and-landing aerial vehicle.

© 2015 Springer Science+Business Media New York Model predictive control (MPC) is a common approach to the control of trajectory-following systems. For nonlinear plants such as car-like robots, methods for path planning and following have the advantage of concurrently solving problems of obstacle avoidance, feasible trajectory selection, and trajectory following. A prediction function for the plant is used to simulate the trajectory with a candidate stream of inputs. Constraints on control inputs and state values, used to ensure safe trajectories and to avoid obstacles, are encoded into a cost function, and optimization routines (at runtime) compute the trajectories and their corresponding control inputs. Such approaches are computationally intensive, and in the nonlinear case the computational burden generally grows as a predictive model more closely approximates a nonlinear plant. In situations where system safety is paramount, guaranteeing model accuracy (in order to achieve more accurate behavior) comes at the cost of increased computation time, which results in increased travel time without a new solution. While the computational burden of predictive methods can be addressed through model reduction, the cost of modeling error over the prediction horizon is high and can lead to unfeasible results. In this paper, we consider the problem of controlling a ground vehicle under constraints and propose an algorithm that employs two models of the vehicle for model predictive control, one coarse and the other more accurate. We introduce a metric called uncontrollable divergence and, using this metric, propose a mechanism to select the model to use in the predictive controller. The novel property of the metric is that it reveals the divergence between predicted and true states caused by return time and model mismatch. More precisely, a map of uncontrollable divergence plotted over the state space gives the criterion to judge where coarse models can be tolerated when a high update rate is preferred (e.g., at high speed and small steering angles), and where high-fidelity models are required to avoid obstacles or make tighter curves (e.g., at large steering angles). With this metric, we design a controller that switches at runtime between predictive controllers in which respective models are deployed. The algorithm is a hybrid controller, which evaluates the proposed metric to select the discrete vehicle model to use for prediction and optimization. We say that the approach is computationally aware, in that the optimization time of each predictive model is dependent on the computation substrate used (chipset, machine architecture, etc.); if a different computational platform is used, then the uncontrollable divergence calculations will lead to a hybrid controller suitable to meet the computation demands for that platform. While the ideas are presented for the solution of a vehicle control problem, the approach has the potential to impact other computationally-demanding cyber-physical systems. The paper extends (Zhang et al., Proceedings of the international conference on cyber-physical system, Seattle, 2015) in a significant way, by demonstrating the calculation of uncontrollable divergence on a physical platform, by characterizing MPC return time as a function of the number of obstacles, and by simulating performance with trajectories that must navigate more obstacles.

The use of nonlinear model-predictive methods for path planning and following has the advantage of concurrently solving problems of obstacle avoidance, feasible trajectory selection, and trajectory following, while obeying constraints 011 control inputs and state values. However, such approaches are computationally intensive, and may not be guaranteed to return a result in bounded time when performing a non- convex optimization. This problem is an interesting application to cyber-physical systems due to their reliance 011 computation to carry out complex control. The computa-tional burden can be addressed through model reduction, at a cost of potential (bounded) model error over the prediction horizon. In this paper we introduce a metric called uncontrollable divergence, and discuss how the selection of the model to use for the predictive controller can be addressed by evaluating this metric, which reveals the divergence between predicted and true states caused by return time and model mismatch. A map of uncontrollable divergence plotted over the state space gives the criterion to judge where reduced models can be tolerated when high update rate is preferred (e.g. at high speed and small steering angles), and where high-fidelity models are required to avoid obstacles or make tighter curves (e.g. at large steering angles). With this metric, we design a hybrid controller that switches at runtime between predictive controllers in which respective models are deployed.

The impact of model-based design on the software engineering community is impressive, and recent research in model transformations, and elegant behavioral specifications of systems has the potential to revolutionize the way in which systems are designed. Such techniques aim to raise the level of abstraction at which systems are specified, to remove the burden of producing application-specific programs with general-purpose programming. For complex real-time systems, however, the impact of model-driven approaches is not nearly so widespread. In this paper, we present a perspective of model-based design researchers who joined with software experts in robotics to enter the DARPA Urban Challenge, and to what extent model-based design techniques were used. Further, we speculate on why, according to our experience and the testimonies of many teams, the full promises of model-based design were not widely realized for the competition. Finally, we present some thoughts for the future of model-based design in complex systems such as these, and what advancements in modeling are needed to motivate small-scale projects to use model-based design in these domains.