Distributed Model Predictive Control for Linear Systems with Adaptive Terminal Sets

In this paper, we propose a distributed model predictive control (DMPC) scheme for linear time-invariant constrained systems which admit a separable structure. To exploit the merits of distributed computation algorithms, the stabilizing terminal controller, value function and invariant terminal set of the DMPC optimization problem need to respect the loosely coupled structure of the system. Although existing methods in the literature address this task, they typically decouple the synthesis of terminal controllers and value functions from the one of terminal sets. In addition, these approaches do not explicitly consider the effect of the current state of the system in the synthesis process. These limitations can lead the resulting DMPC scheme to poor performance since it may admit small or even empty terminal sets. Unlike other approaches, this paper presents a unified framework to encapsulate the synthesis of both the stabilizing terminal controller and invariant terminal set into the DMPC formulation. Conditions for Lyapunov stability and invariance are imposed in the synthesis problem in a way that allows the value function and invariant terminal set to admit the desired distributed structure. We illustrate the effectiveness of the proposed method on several examples including a benchmark spring-mass-damper problem

Sample Complexity of Sparse System Identification Problem

In this paper, we study the system identification problem for sparse linear time-invariant systems. We propose a sparsity promoting block-regularized estimator to identify the dynamics of the system with only a limited number of input-state data samples. We characterize the properties of this estimator under high-dimensional scaling, where the growth rate of the system dimension is comparable to or even faster than that of the number of available sample trajectories. In particular, using contemporary results on high-dimensional statistics, we show that the proposed estimator results in a small element-wise error, provided that the number of sample trajectories is above a threshold. This threshold depends polynomially on the size of each block and the number of nonzero elements at different rows of input and state matrices, but only logarithmically on the system dimension. A by-product of this result is that the number of sample trajectories required for sparse system identification is significantly smaller than the dimension of the system. Furthermore, we show that, unlike the recently celebrated least-squares estimators for system identification problems, the method developed in this work is capable of \textit{exact recovery} of the underlying sparsity structure of the system with the aforementioned number of data samples. Extensive case studies on synthetically generated systems, physical mass-spring networks, and multi-agent systems are offered to demonstrate the effectiveness of the proposed method