Neural System Level Synthesis: Learning over All Stabilizing Policies for Nonlinear Systems

Abstract

The growing scale and complexity of safety-critical control systems underscore the need to evolve current control architectures aiming for the unparalleled performances achievable through state-of-the-art optimization and machine learning algorithms. However, maintaining closed-loop stability while boosting the performance of nonlinear control systems using data-driven and deep-learning approaches stands as an important unsolved challenge. In this paper, we tackle the performance-boosting problem with closed-loop stability guarantees. Specifically, we establish a synergy between the Internal Model Control (IMC) principle for nonlinear systems and stateof-the-art unconstrained optimization approaches for learning stable dynamics. Our methods enable learning over arbitrarily deep neural network classes of performance-boosting controllers for stable nonlinear systems; crucially, we guarantee Lp closedloop stability even if optimization is halted prematurely, and even when the ground-truth dynamics are unknown, with vanishing conservatism in the class of stabilizing policies as the model uncertainty is reduced to zero. We discuss the implementation details of the proposed control schemes, including distributed ones, along with the corresponding optimization procedures, demonstrating the potential of freely shaping the cost functions through several numerical experiments.

Universal Approximation Property of Hamiltonian Deep Neural Networks

“…However, this approach is limited to quadratic storage functions for subsystems, constraining the flexibility and generalization. Although [3] presented a similar distributed NN framework based on pH systems that ensure passivity by design but not a finite L 2 gain for the closed-loop system which is instead our main result. Unlike passivity, a finite L 2 gain guarantees stability even in the presence of external disturbances or modeling errors, which is crucial for safe operation in uncertain environments [23], [38].…”

“…The next Section presents a novel method that tackles this challenge effectively. Remark 1 (Passivity by design): While achieving passivity by design for the closed-loop system is considered in [3], it may not always ensure stability, especially when controlled system interacts with a passive, but else completely unknown environment. In fact, the converse of the passivity theorem tells us that the controlled system must be output strictly passive as seen from the interaction port of the controlled system with the environment [38].…”

“…Remark 4 (Communication among sub-controllers): Note that our framework can seamlessly incorporate communication graphs among the sub-controllers without loss of generality. In fact, the work [3] provides a systematic approach to interconnect sub-controllers while preserving the dissipativity of the closed-loop. For specific details and an example, we defer the readers to [3,Theorem 3].…”

“…A stream of works such as [2] has provided necessary and sufficient condition, namely, Quadratic Invariance (QI), under which the distributed optimal controller is linear and corresponds to solving a convex optimization problem. However, realworld systems often violate QI assumptions due to inherent nonlinearities, non-convex control costs, or privacy limitations [3]. This necessitates venturing beyond linear control and exploring highly nonlinear distributed policies, such as those parametrized by Deep Neural Networks (DNNs).…”

“…DNNs have proved their capabilities in learning-enabled control [3]- [6], and system identification [7]- [15] of nonlinear dynamical systems. Indeed, NN control has been applied in diverse application domains, such as robotics [4], epidemic models [16], safe path planning [6], and Kuramoto oscillators [17].…”