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Robust offset-free constrained Model Predictive Control with Long Short-Term Memory Networks -- Extended version

Irene Schimperna, Lalo Magni

TL;DR

A control scheme is developed, based on the use of long short-term memory neural network models and nonlinear model predictive control, which guarantees recursive feasibility with slow time variant set-points and disturbances, input and output constraints and unmeasurable state.

Abstract

This paper develops a control scheme, based on the use of Long Short-Term Memory neural network models and Nonlinear Model Predictive Control, which guarantees recursive feasibility with slow time variant set-points and disturbances, input and output constraints and unmeasurable state. Moreover, if the set-point and the disturbance are asymptotically constant, offset-free tracking is guaranteed. Offset-free tracking is obtained by augmenting the model with a disturbance, to be estimated together with the states of the Long Short-Term Memory network model by a properly designed observer. Satisfaction of the output constraints in presence of observer estimation error, time variant set-points and disturbances is obtained using a constraint tightening approach.

Robust offset-free constrained Model Predictive Control with Long Short-Term Memory Networks -- Extended version

TL;DR

A control scheme is developed, based on the use of long short-term memory neural network models and nonlinear model predictive control, which guarantees recursive feasibility with slow time variant set-points and disturbances, input and output constraints and unmeasurable state.

Abstract

This paper develops a control scheme, based on the use of Long Short-Term Memory neural network models and Nonlinear Model Predictive Control, which guarantees recursive feasibility with slow time variant set-points and disturbances, input and output constraints and unmeasurable state. Moreover, if the set-point and the disturbance are asymptotically constant, offset-free tracking is guaranteed. Offset-free tracking is obtained by augmenting the model with a disturbance, to be estimated together with the states of the Long Short-Term Memory network model by a properly designed observer. Satisfaction of the output constraints in presence of observer estimation error, time variant set-points and disturbances is obtained using a constraint tightening approach.
Paper Structure (29 sections, 11 theorems, 176 equations, 8 figures, 1 table)

This paper contains 29 sections, 11 theorems, 176 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notation
  3. Definitions
  4. Problem formulation and control algorithm
  5. Long Short-Term Memory neural network model
  6. LSTM stability properties
  7. Incremental Lyapunov function for the LSTM
  8. State and disturbance observer
  9. Disturbance model
  10. Observer design
  11. Observer convergence
  12. Reference calculator
  13. Robust MPC formulation
  14. Stability and offset-free results
  15. Recursive feasibility and stability analysis
  16. ...and 14 more sections

Key Result

Theorem 1

Let Assumption ass:delta_iss_condition hold. Then it is possible to upper bound the difference between any couple of system trajectories $x_{\mathrm{a}} = [c_{\mathrm{a}}^\top \; h_{\mathrm{a}}^\top]^\top$ and $x_{\mathrm{b}} = [c_{\mathrm{b}}^\top \; h_{\mathrm{b}}^\top]^\top$ with the following in where and the LSTM system eq:lstm is exponentially $\delta$ISS in the sets $\mathcal{X}$ and $\mat

Figures (8)

  • Figure 1: Block diagram of the control scheme.
  • Figure 2: Block diagram of the nominal closed-loop control scheme, where the real plant is replaced by its LSTM model augmented with a disturbance term.
  • Figure 3: Schematic layout of the pH neutralization process.
  • Figure 4: Evolution of the output of the LSTM model (blue line) compared with reference $y^0$ (black dashed line), and evolution of the disturbance $d$ (orange dash-dotted line). $y^0$ and $d$ are selected to respect the sufficient conditions for recursive feasibility of Theorem \ref{['th:feasibility_stability']}.
  • Figure 5: Evolution of the output of the LSTM model (blue line) compared with reference $y^0$ (black dashed line), and evolution of the disturbance $d$ (orange dash-dotted line). The sufficient conditions for recursive feasibility of Theorem \ref{['th:feasibility_stability']} are not respected, but recursive feasibility is maintained in practice.
  • ...and 3 more figures

Theorems & Definitions (20)

  • Remark 1
  • Theorem 1: terzi2021mpc_lstm
  • Remark 2
  • Lemma 1
  • Lemma 2
  • Corollary 1
  • Remark 3
  • Remark 4
  • Lemma 3
  • Theorem 2
  • ...and 10 more