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Development of a velocity form for a class of RNNs, with application to offset-free nonlinear MPC design

Daniele Ravasio, Bestem Abdulaziz, Marcello Farina, Andrea Ballarino

TL;DR

This paper tackles offset-free tracking for nonlinear plants modeled by deep RNNs, proposing a velocity-form reformulation that recasts tracking as regulation and enables stability guarantees via LMIs. It develops a complete control pipeline: a nonlinear state-feedback law with LMI-based design to guarantee global or regional stability, an offset-free NMPC that uses the velocity form as the predictor and incorporates terminal components, and a state observer for disturbances and model-plant mismatches. The key contributions are the velocity-form extension to deep RNNs, an incremental sector condition-driven LMI design for both the controller and observer, and a theoretically sound NMPC framework that enlarges the region of attraction while enforcing input-output constraints; demonstrated on a pH-neutralisation benchmark with robust tracking under disturbances. Together, these results provide a principled, data-driven control methodology for complex nonlinear dynamics with provable stability and constraint handling in practical process settings.

Abstract

This paper addresses the offset-free tracking problem for nonlinear systems described by a class of recurrent neural networks (RNNs). To compensate for constant disturbances and guarantee offset-free tracking in the presence of model-plant mismatches, we propose a novel reformulation of the RNN model in velocity form. Conditions based on linear matrix inequalities are then derived for the design of a nonlinear state observer and a nonlinear state-feedback controller, ensuring global or regional closed-loop stability of the origin of the velocity form dynamics. Moreover, to handle input and output constraints, a theoretically sound offset-free nonlinear model predictive control algorithm is developed. The algorithm exploits the velocity form model as the prediction model and the static controller as an auxiliary law for the definition of the terminal ingredients. Simulations on a pH-neutralisation process benchmark demonstrate the effectiveness of the proposed approach.

Development of a velocity form for a class of RNNs, with application to offset-free nonlinear MPC design

TL;DR

This paper tackles offset-free tracking for nonlinear plants modeled by deep RNNs, proposing a velocity-form reformulation that recasts tracking as regulation and enables stability guarantees via LMIs. It develops a complete control pipeline: a nonlinear state-feedback law with LMI-based design to guarantee global or regional stability, an offset-free NMPC that uses the velocity form as the predictor and incorporates terminal components, and a state observer for disturbances and model-plant mismatches. The key contributions are the velocity-form extension to deep RNNs, an incremental sector condition-driven LMI design for both the controller and observer, and a theoretically sound NMPC framework that enlarges the region of attraction while enforcing input-output constraints; demonstrated on a pH-neutralisation benchmark with robust tracking under disturbances. Together, these results provide a principled, data-driven control methodology for complex nonlinear dynamics with provable stability and constraint handling in practical process settings.

Abstract

This paper addresses the offset-free tracking problem for nonlinear systems described by a class of recurrent neural networks (RNNs). To compensate for constant disturbances and guarantee offset-free tracking in the presence of model-plant mismatches, we propose a novel reformulation of the RNN model in velocity form. Conditions based on linear matrix inequalities are then derived for the design of a nonlinear state observer and a nonlinear state-feedback controller, ensuring global or regional closed-loop stability of the origin of the velocity form dynamics. Moreover, to handle input and output constraints, a theoretically sound offset-free nonlinear model predictive control algorithm is developed. The algorithm exploits the velocity form model as the prediction model and the static controller as an auxiliary law for the definition of the terminal ingredients. Simulations on a pH-neutralisation process benchmark demonstrate the effectiveness of the proposed approach.

Paper Structure

This paper contains 17 sections, 6 theorems, 86 equations, 3 figures.

Table of Contents

  1. Introduction
  2. PROBLEM STATEMENT
  3. The plant model
  4. The velocity form
  5. Incremental sector condition
  6. Relationship between velocity form states and states of the original model
  7. Control design
  8. Static state-feedback control law
  9. Design procedure
  10. MPC control design
  11. Velocity form NMPC design
  12. Terminal ingredients
  13. Main result
  14. State observer design
  15. Case study
  16. ...and 2 more sections

Key Result

Lemma 1

Consider a square matrix $E \in \mathbb{R}^{n \times n}$. If there exists a matrix $P \in \mathbb{D}_+^n$ such that then, $E \in \mathbb{B}_\Theta$. $\square$

Figures (3)

  • Figure 1: pH-neutralization process.
  • Figure 2: Closed-loop output performance. Black dashed lines denote output constraints.
  • Figure 3: Evolution of the control input. Black dashed lines denote input constraints.

Theorems & Definitions (6)

  • Lemma 1
  • Lemma 2
  • Proposition 3
  • Theorem 4
  • Theorem 5
  • Theorem 6