An Incremental Input-to-State Stability Condition for a Class of Recurrent Neural Networks

Abstract

This paper proposes a novel sufficient condition for the incremental input-to-state stability of a class of recurrent neural networks (RNNs). The established condition is compared with others available in the literature, showing to be less conservative. Moreover, it can be applied for the design of incremental input-to-state stable RNN-based control systems, resulting in a linear matrix inequality constraint for some specific RNN architectures. The formulation of nonlinear observers for the considered system class, as well as the design of control schemes with explicit integral action, are also investigated. The theoretical results are validated through simulation on a referenced nonlinear system.

Recurrent Neural Network-Based MPC for Systems With Input and Incremental Input Constraints

“…□ δISS conditions and incremental Lyapunov functions that satisfy (3) can be found for the main RNN architectures (LSTM [8], GRU [9], [12], REN [15], and other particular classes of RNN [10]). Moreover, the property of local Lipschitzianity is satisfied by quadratic functions, that are the most common types of Lyapunov functions used to study RNN stability.…”

“…for any input sequence in U, ∥x k − xk ∥ → 0 for k → ∞. □ Observers respecting Assumption 2 have been designed for the main RNN architectures (LSTM [8], GRU [13], REN [15] and other particular classes of RNN [10]).…”

“…In order to guarantee that the effect of the state initialization vanishes, and that a small input variation does not lead to a large state variation, it is useful to consider RNN models that satisfy an Incremental Input-to-State Stability (δISS) property [7]. Sufficient conditions for δISS have been derived for the most common classes of RNN, such as for Long Short-Term Memory (LSTM) in [8], for Gated Recurrent Units (GRU) in [9] and other particular classes of RNN including Echo State Networks (ESN) and Neural Nonlinear Autoregressive eXogenous models (NNARX) in [10]. For all these networks, a straightforward modification of the training loss can lead to a final model that satisfies the desired δISS condition.…”