Sampled-Data Online Feedback Equilibrium Seeking: Stability and Tracking

Abstract

This paper proposes a general framework for constructing feedback controllers that drive complex dynamical systems to "efficient" steady-state (or slowly varying) operating points. Efficiency is encoded using generalized equations which can model a broad spectrum of useful objectives, such as optimality or equilibria (e.g. Nash, Wardrop, etc.) in noncooperative games. The core idea of the proposed approach is to directly implement iterative solution (or equilibrium seeking) algorithms in closed loop with physical systems. Sufficient conditions for closed-loop stability and robustness are derived; these also serve as the first closed-loop stability results for sampleddata feedback-based optimization. Numerical simulations of smart building automation and game-theoretic robotic swarm coordination support the theoretical results.

Tutorial on Congestion Control in Multi-Area Transmission Grids via Online Feedback Equilibrium Seeking

“…Multiple iterations (mostly inspired by iterative algorithms in nonlinear optimization) serve this purpose, including • (projected) gradient iterations [6], [31] • primal-dual saddle-point dynamics [6], [9], [32] • safe gradient flows [33] • regularized primal-dual iterations [10] • quasi-Newton flows [34] • sequential convex programming [22] • and others. We refer to [22], for a general framework encompassing most of the algorithms listed above. An important difference between these iterations is how they handle the constraints of the problem, and in particular the output constraints (2d).…”

“…Game theory, which studies dynamics of conflict and cooperation between self-interested rational decision makers, offers a powerful framework to approach this question. The development of game-theoretic controllers for economic steady-state regulation of complex systems has been recently studied in different works [21]- [24]. All these works focus on the design of game-theoretic feedback controllers, but the technical tools therein can be also used for analysis purposes.…”

Robust Laguerre‐based model predictive control for trajectory tracking of a mobile robot using an linear matrix inequality (LMI)‐based approach

“…In multiagent systems, for instance, large-scale optimal control should be solved in real-time. Generally, such algorithms require multiple communication cycles between subsystems to converge, which is a significant drawback from a computational and energy standpoint (for wireless systems) and quickly becomes the performance bottleneck [23,24]. Consequently, this proposed idea in this paper and other research in the field of suboptimal controllers can be immensely beneficial for complex systems.…”

“…Consequently, this proposed idea in this paper and other research in the field of suboptimal controllers can be immensely beneficial for complex systems. For example, in [23], a type of suboptimal MPC is used to alleviate this burden for a complex and constrained multiagent system, and to reduce the computational complexity of MPC, the authors of [25] implemented a Laguerre-based MPC with a tube-based scheme to track a trajectory by a mobile robot in both the LTI and LTV cases. In this manner, robustness is achieved, constraints are held, and stability is guaranteed.…”

Novel Control Approaches Based on Projection Dynamics

“…Furthermore, they also establish convergence by employing Lyapunov theory and the properties of the projection operator. With the advent of distributed systems, the focus shifted to combining the projection dynamics with distributed optimization [12] and control [13], [14], [15].…”