The modular open-source framework GRAMPC-D for model predictive control of distributed systems is presented in this paper. The modular concept allows to solve optimal control problems in a centralized and distributed fashion using the same problem description. It is tailored to computational efficiency with the focus on embedded hardware. The distributed solution is based on the alternating direction method of multipliers and uses the concept of neighbor approximation to enhance convergence speed. The presented framework can be accessed through C++ and Python and also supports plug-and-play and data exchange between agents over a network.

The major part of the execution time of distributed algorithms is required for the communication between agents. This paper approaches a reduction of the communication effort by reducing the number of edges in the considered graph. This is achieved by partitioning the graph and formulating a super graph. At first, the computational and communication effort is evaluated on an abstract level independent of the distributed algorithm, before the Alternating Direction Method of Multipliers (ADMM) is applied to a system of coupled water tanks. This allows to outline the trade-off between computation and communication time and to evaluate an optimal number of partitions that minimizes the execution time. The influence of the partitioning on the convergence behavior of the distributed algorithm is studied and compared with the concept of neighbor approximation.

Model predictive control (MPC) has widely been used in both research and application in recent years. Methods from the field of distributed MPC (DMPC) are especially designed to handle large systems or applications with demands in terms of flexibility and modularity. Two distinctive algorithms for DMPC, namely Sensitivity-based DMPC (SENSI) and the Alternating Directions Method of Multipliers (ADMM) are compared in this paper concerning their applicability to building automation. First, the algorithms are presented and applied to two academic examples in order to analyze the convergence behavior and scalability of the algorithms. It is shown that ADMM and SENSI algorithm maintain constant computation time per agent with increasing number of agents in the network. Afterwards, the algorithms are applied to a building automation system in a setpoint control and an energy optimal control scenario. Both algorithms are capable of solving the corresponding optimal control problems (OCP) which demonstrates the applicability of the algorithms to building automation. Furthermore, the differences of the algorithms in scope of this application are analyzed showing a better performance of the ADMM algorithm in case of setpoint control. However, the sensitivity-based approach needs less communication steps than the ADMM in an energy optimal control scenario.

This paper presents an asynchronous formulation of the alternating direction method of multipliers (ADMM) for solving optimal control problems that arise in distributed model predictive control (DMPC). The need for synchronization limits the applicability of DMPC to large-scale or fast nonlinear systems, since all agents have to wait for the slowest one. The main idea of the asynchronous formulation is to use data from previous iterations instead of waiting for the current data. A heuristic is proposed to adapt the maximum allowed delay depending on the residuals of the consistency constraints. The convergence behaviour is investigated in a numerical simulation and the execution time is evaluated using distributed hardware and TCP communication. The results show that the reduced execution time more than compensates for the disadvantage of slower convergence compared to the synchronous formulation.

The open-source framework GRAMPC-D enables controlling systems based on distributed model predictive control (DMPC) by providing an implementation that is capable of solving the Alternating Direction Method of Multipliers algorithm (ADMM) in the millisecond range. It supports communication over a network and plug-and-play mechanisms. This paper presents an experimental validation of GRAMPC-D applied to the multi-agent laboratory Robotarium. In the first experiment, a distribution problem is considered, whereby four robots maximize the distance between them. A formation control scenario is evaluated in the second scenario, whereby eleven robots form a rotating letter “G”. In the third experiment, a platooning scenario is evaluated in combination with plug-and-play mechanisms. The per-agent computation time in each experiment is lower than the sampling time. As these scenarios would exceed the system class that is currently considered in the implementation of GRAMPC-D, it is extended to consider agents that are coupled in their cost functional.

This paper concerns the distributed solution of optimal control problems for the class of nonlinear neighbor-affine systems, for which the dynamics can be written as sum of terms that depend only on the states and controls of two neighboring agents. The proposed distributed optimization algorithm is based on the introduction of local copies of states and controls that are coupled via additional consistency constraints. Formulating the augmented Lagrangian function for the consistency constraints then allows to solve the problem in a distributed manner using the alternating direction method of multipliers (ADMM). This paper proposes to approximate parts of the neighbors optimization problem using local copies of states and controls. While this increases the complexity of the local problems, an extensive numerical evaluation for several benchmark examples shows that the convergence behaviour of the ADMM algorithm is improved significantly and thus the total computation time and the communication effort are decreased.

This paper extends the recently introduced ALADIN algorithm to non-convex continuous-time optimal control problems with nonlinear dynamics and linear coupling constraints. The algorithm alternates between solving a convexified local problem in a distributed manner and a linearized quadratic problem on a centralized entity while using the solution of both for an update step. This paper presents an analysis of the local convergence of the algorithm and shows a quadratic convergence rate. Furthermore, a globalization strategy is presented that ensures global convergence. The paper closes with a numerical evaluation of the algorithm.

The development of real-time capable distributed model predictive controllers (DMPC) for nonlinear systems requires considerable effort both for the numerical solution and for the practical implementation, which is one of the reasons why DMPC so far is rarely found in practice. With the aim of simplifying the development process, this paper presents a modular framework for DMPC of nonlinear neighbor-affine systems. The numerical solution is based on the alternating direction method of multipliers (ADMM). The local optimization problems are solved using the toolbox GRAMPC which implements a gradient-based augmented Lagrangian algorithm that is particularly suited for embedded applications. The performance of the DMPC framework is evaluated for two scalable nonlinear distributed systems, whereby the scalability and the plug-and-play functionality are demonstrated.