Recurrent neural network-based robust control systems with closed-loop regional incremental ISS and application to MPC design

TL;DR

This work tackles robust output-feedback control for nonlinear plants modeled by recurrent neural networks by introducing an LMI-based observer and a static state-feedback law that enforce $\delta$ISS$ (\delta$ISS) regionally or globally to track constant setpoints under disturbances. To mitigate conservatism from global properties, it leverages a novel generalized incremental sector condition and provides an alternative tube-based NMPC design that enlarges the feasible region while guaranteeing recursive feasibility. The approach is validated on a pH-neutralisation benchmark using an $8$-state RNN with sigmoid/tanh activations, demonstrating that the NMPC variant significantly enlarges the region of attraction compared with the static law while maintaining tracking accuracy and constraint satisfaction. Computationally, the LMIs scale polynomially with system size, offering a practical method for nonlinear control of RNN-based models in moderate dimensions and potentially enabling extensions to larger or deeper networks with data-driven disturbance modeling.

Abstract

This paper investigates the design of output-feedback schemes for systems described by a class of recurrent neural networks. We propose a procedure based on linear matrix inequalities for designing an observer and a static state-feedback controller. The algorithm leverages global and regional incremental input-to-state stability (incremental ISS) and enables the tracking of constant setpoints, ensuring robustness to disturbances and state estimation uncertainty. To address the potential limitations of regional incremental ISS, we introduce an alternative scheme in which the static law is replaced with a tube-based nonlinear model predictive controller (NMPC) that exploits regional incremental ISS properties. We show that these conditions enable the formulation of a robust NMPC law with guarantees of convergence and recursive feasibility, leading to an enlarged region of attraction. Theoretical results are validated through numerical simulations on the pH-neutralisation process benchmark.

Figures (5)

• Figure 1: Output-feedback control architecture with static control law.
• Figure 2: NMPC output-feedback control architecture.
• Figure 3: Range of initial conditions expressed in terms of the output $y$ from which $\bar{y}$ is reachable. The figure compares the static control law and the NMPC law with $N=3$ and $N=10$.
• Figure 4: Closed-loop output performance. Black dashed lines denote output constraints. Sshaded regions represent tubes around the nominal trajectories.
• Figure 5: Evolution of the control input. Black dashed lines denote input constraints. Shaded regions represent tubes around the nominal trajectories.

Theorems & Definitions (15)

• Definition 1: $\delta$ISS, bayer2013discrete
• Definition 2: Dissipation-form $\delta$ISS Lyapunov function, bayer2013discrete
• Theorem 1: bayer2013discrete
• Definition 3: $\delta$ISS observer error dynamics
• Lemma 2
• Proposition 3
• Lemma 4
• Proposition 5
• Lemma 6
• Theorem 7