Economic MPC of Nonlinear Systems with Non-Monotonic Lyapunov Functions and Its Application to HVAC Control

This paper proposes a Lyapunov-based economic MPC scheme for nonlinear sytems with non-monotonic Lyapunov functions. Relaxed Lyapunov-based constraints are used in the MPC formulation to improve the economic performance. These constraints will enforce a Lyapunov decrease after every few steps. Recursive feasibility and asymptotical convergence to the steady state can be achieved using Lyapunov-like stability analysis. The proposed economic MPC can be applied to minimize energy consumption in HVAC control of commercial buildings. The Lyapunov-based constraints in the online MPC problem enable the tracking of the desired set-point temperature. The performance is demonstrated by a virtual building composed of two adjacent zones

Fully Distributed Nash Equilibrium Seeking in N-Cluster Games

Distributed optimization and Nash equilibrium (NE) seeking problems have drawn much attention in the control community recently. This paper studies a class of non-cooperative games, known as $N$-cluster game, which subsumes both cooperative and non-cooperative nature among multiple agents in the two problems: solving distributed optimization problem within the cluster, while playing a non-cooperative game across the clusters. Moreover, we consider a partial-decision information game setup, i.e., the agents do not have direct access to other agents' decisions, and hence need to communicate with each other through a directed graph whose associated adjacency matrix is assumed to be non-doubly stochastic. To solve the $N$-cluster game problem, we propose a fully distributed NE seeking algorithm by a synthesis of leader-following consensus and gradient tracking, where the leader-following consensus protocol is adopted to estimate the other agents' decisions and the gradient tracking method is employed to trace some weighted average of the gradient. Furthermore, the algorithm is equipped with uncoordinated constant step-sizes, which allows the agents to choose their own preferred step-sizes, instead of a uniform coordinated step-size. We prove that all agents' decisions converge linearly to their corresponding NE so long as the largest step-size and the heterogeneity of the step-size are small. We verify the derived results through a numerical example in a Cournot competition game

Gradient-Free Nash Equilibrium Seeking in N-Cluster Games with Uncoordinated Constant Step-Sizes

In this paper, we consider a problem of simultaneous global cost minimization and Nash equilibrium seeking, which commonly exists in $N$-cluster non-cooperative games. Specifically, the agents in the same cluster collaborate to minimize a global cost function, being a summation of their individual cost functions, and jointly play a non-cooperative game with other clusters as players. For the problem settings, we suppose that the explicit analytical expressions of the agents' local cost functions are unknown, but the function values can be measured. We propose a gradient-free Nash equilibrium seeking algorithm by a synthesis of Gaussian smoothing techniques and gradient tracking. Furthermore, instead of using the uniform coordinated step-size, we allow the agents across different clusters to choose different constant step-sizes. When the largest step-size is sufficiently small, we prove a linear convergence of the agents' actions to a neighborhood of the unique Nash equilibrium under a strongly monotone game mapping condition, with the error gap being propotional to the largest step-size and the smoothing parameter. The performance of the proposed algorithm is validated by numerical simulations