Gradient-Based Distributed Model Predictive Control

Abstract: The thesis covers different topics related to model predictive control (MPC) and particularly distributed model predictive control (DMPC). One topic of the thesis is gradient-based optimization algorithms for solving the optimization problem arising in DMPC in a distributed manner. The underlying idea is to solve the optimization problem in distributed fashion using dual decomposition, which is a well-known method. Dual decomposition is traditionally used in conjunction with (sub)gradient methods which are known to have bad convergence rate properties, especially for ill-conditioned problem. In this thesis it is shown how to use accelerated gradient methods with dual decomposition, and how to choose the step size parameter optimally in the algorithm. A method to bound the number of iterations needed to guarantee a prespecified accuracy of the solution is also provided. Based on the iteration bound, it is shown how to precondition the problem data optimally to improve conditioning of the problem. These contributions significantly improve the performance of the distributed optimization algorithm compared to dual decomposition with a (sub)gradient method.

Another topic of the thesis is to guarantee feasibility and stability when using the developed distributed optimization algorithm in a DMPC context. Traditional methods of proving stability in MPC usually involve terminal cost functions and terminal constraints that are non-separable. These methods are not directly applicable in DMPC based on dual decomposition because of the non-separable terms. Further, dual decomposition does not provide feasible iterations but is guaranteed to be primal feasible only in the limit. These issues have been addressed in the thesis. The stability issue is addressed by showing that for problems without a terminal cost or terminal constraints and if a certain controllability assumption on the stage costs is satisfied, the optimal value function is decreasing in every time step by a prespecified amount. It is also shown how the controllability assumption can be verified by solving a mixed integer linear program. The feasibility issue is addressed by a novel adaptive constraint tightening approach. The adaptive constraint tightening guarantees that a primal feasible solution can be constructed with finite number of algorithm iterations without compromising the stability guarantee.

The developed distributed optimization algorithm is evaluated on a hydro power valley benchmark problem. The hydro power valley consists of several dams connected in series where each dam is equipped with a turbine to extract power from the water. The objective is to control the water flow between the dams such that the total power from the turbines matches a power reference while respecting constraints on water levels and water flows. The control problem is formulated as an optimization problem, which is solved in receding horizon fashion using the distributed optimization algorithm presented in the thesis. The performance of the proposed distributed controller is compared to the performance of a centralized controller.