Riccardo Scattolini

Enrico Terzi, Lorenzo Fagiano, Marcello Farina et al.

A comprehensive approach addressing identification and control for learningbased Model Predictive Control (MPC) for linear systems is presented. The design technique yields a data-driven MPC law, based on a dataset collected from the working plant. The method is indirect, i.e. it relies on a model learning phase and a model-based control design one, devised in an integrated manner. In the model learning phase, a twofold outcome is achieved: first, different optimal p-steps ahead prediction models are obtained, to be used in the MPC cost function; secondly, a perturbed state-space model is derived, to be used for robust constraint satisfaction. Resorting to Set Membership techniques, a characterization of the bounded model uncertainties is obtained, which is a key feature for a successful application of the robust control algorithm. In the control design phase, a robust MPC law is proposed, able to track piece-wise constant reference signals, with guaranteed recursive feasibility and convergence properties. The controller embeds multistep predictors in the cost function, it ensures robust constraints satisfaction thanks to the learnt uncertainty model, and it can deal with possibly unfeasible reference values. The proposed approach is finally tested in a numerical example.

Fabio Bonassi, Alessio La Bella, Marcello Farina et al.

This brief addresses the design of a Nonlinear Model Predictive Control (NMPC) strategy for exponentially incremental Input-to-State Stable (ISS) systems. In particular, a novel formulation is devised, which does not necessitate the onerous computation of terminal ingredients, but rather relies on the explicit definition of a minimum prediction horizon ensuring closed-loop stability. The designed methodology is particularly suited for the control of systems learned by Recurrent Neural Networks (RNNs), which are known for their enhanced modeling capabilities and for which the incremental ISS properties can be studied thanks to simple algebraic conditions. The approach is applied to Gated Recurrent Unit (GRU) networks, providing also a method for the design of a tailored state observer with convergence guarantees. The resulting control architecture is tested on a benchmark system, demonstrating its good control performances and efficient applicability.

Luca Bugliari Armenio, Enrico Terzi, Marcello Farina et al.

The goal of this paper is to investigate the theoretical properties, the training algorithm, and the predictive control applications of Echo State Networks (ESNs), a particular kind of Recurrent Neural Networks. First, a condition guaranteeing incremetal global asymptotic stability is devised. Then, a modified training algorithm allowing for dimensionality reduction of ESNs is presented. Eventually, a model predictive controller is designed to solve the tracking problem, relying on ESNs as the model of the system. Numerical results concerning the predictive control of a nonlinear process for pH neutralization confirm the effectiveness of the proposed algorithms for the identification, dimensionality reduction, and the control design for ESNs.

Marcello Farina, Luca Giulioni, Lalo Magni et al.

In this paper we propose an output-feedback Model Predictive Control (MPC) algorithm for linear discrete-time systems affected by a possibly unbounded additive noise and subject to probabilistic constraints. In case the noise distribution is unknown, the chance constraints on the input and state variables are reformulated by means of the Chebyshev - Cantelli inequality. The recursive feasibility of the proposed algorithm is guaranteed and the convergence of the state to a suitable neighbor of the origin is proved under mild assumptions. The implementation issues are thoroughly addressed showing that, with a proper choice of the design parameters, its computational load can be made similar to the one of a standard stabilizing MPC algorithm. Two examples are discussed in details, with the aim of providing an insight on the performance achievable by the proposed control scheme.

Fabio Bonassi, Jing Xie, Marcello Farina et al.

This paper proposes a method for lifelong learning of Recurrent Neural Networks, such as NNARX, ESN, LSTM, and GRU, to be used as plant models in control system synthesis. The problem is significant because in many practical applications it is required to adapt the model when new information is available and/or the system undergoes changes, without the need to store an increasing amount of data as time proceeds. Indeed, in this context, many problems arise, such as the well known Catastrophic Forgetting and Capacity Saturation ones. We propose an adaptation algorithm inspired by Moving Horizon Estimators, deriving conditions for its convergence. The described method is applied to a simulated chemical plant, already adopted as a challenging benchmark in the existing literature. The main results achieved are discussed.

Fabio Bonassi, Alessio La Bella, Giulio Panzani et al.

The aim of this work is to investigate the use of Incrementally Input-to-State Stable ($δ$ISS) deep Long Short Term Memory networks (LSTMs) for the identification of nonlinear dynamical systems. We show that suitable sufficient conditions on the weights of the network can be leveraged to setup a training procedure able to learn provenly-$δ$ISS LSTM models from data. The proposed approach is tested on a real brake-by-wire apparatus to identify a model of the system from input-output experimentally collected data. Results show satisfactory modeling performances.

Marcello Farina, Xinglong Zhang, Riccardo Scattolini

This paper describes a hierarchical control scheme for interconnected systems. The higher layer of the control structure is designed with robust Model Predictive Control (MPC) based on a reduced order dynamic model of the overall system and is aimed at optimizing long-term performance, while at the lower layer local regulators acting at a higher frequency are designed for the full order models of the subsystems to refine the control action. A simulation experiment concerning the control of the temperature inside a building is reported to witness the potentialities of the proposed approach.

Laura Boca de Giuli, Alessio La Bella, Manish Prajapat et al.

A key challenge in learning-based model predictive control (MPC) is to collect informative data online for model adaptation while ensuring safety and without penalising control performance. In this paper, we propose an online model adaptation scheme embedded within an MPC framework in which the last-layer parameters of a recurrent neural network are recursively updated via Bayesian learning. This is achieved by means of a goal-oriented safe active learning algorithm that alternates between an exploration phase, where the MPC actively explores system dynamics to collect informative data for model adaptation while still pursuing the main control objective, and a goal-reaching phase, where it focuses exclusively on the main control objective. The algorithm is complemented with theoretical guarantees of (i) recursive feasibility, (ii) safety, (iii) termination of exploration in finite time, and (iv) close-to-optimal performance. Simulation results on a benchmark energy system demonstrate that the proposed framework achieves economic performance comparable to that of an MPC with full system knowledge, while progressively improving model accuracy and respecting operational safety constraints with high probability.

Enrico Terzi, Lorenzo Fagiano, Marcello Farina et al.

The derivation of multi-step-ahead prediction models from sampled data of a linear system is considered. A dedicated prediction model is built for each future time step of interest. In addition to a nominal model, the set of all models consistent with data and prior information is derived as well, making the approach suitable for robust control design within a Model Predictive Control framework. The resulting parameter identification problem is solved through a sequence of convex programs, overcoming the non-convexity arising when identifying 1-step prediction models with an output-error criterion. At the same time, the derived models guarantee a worst-case error which is always smaller than the one obtained by iterating models identified with a 1-step prediction error criterion.

Laura Boca de Giuli, Samuel Mallick, Alessio La Bella et al.

This paper presents a model predictive control (MPC) framework leveraging an ensemble of data-based models to optimally control complex systems under multiple operating conditions. A novel combination rule for ensemble models is proposed, based on the statistical Mahalanobis distance, enabling the ensemble weights to suitably vary across the prediction window based on the system input. In addition, a novel state observer for ensemble models is developed using moving horizon estimation (MHE). The effectiveness of the proposed methodology is demonstrated on a benchmark energy system operating under multiple conditions.

Samuel Mallick, Laura Boca de de Giuli, Alessio La Bella et al.

This paper addresses the design of an event-triggered, data-based, and performance-oriented adaption method for model predictive control (MPC). The performance of such a strategy strongly depends on the accuracy of the prediction model, which may require online adaption to prevent performance degradation under changing operating conditions. Unlike existing methods that continuously update model and control parameters from data, potentially leading to catastrophic forgetting and unnecessary control modifications, we propose a novel approach based on statistical monitoring of closed-loop performance indicators. This framework enables the detection of performance degradation, and, when required, controller adaption is performed via reinforcement learning and identification techniques. The proposed strategy is validated on a high-fidelity simulation of a district heating system benchmark.

Fabio Bonassi, Jing Xie, Marcello Farina et al.

This paper deals with the design of nonlinear MPC controllers that provide offset-free setpoint tracking for models described by Neural Nonlinear AutoRegressive eXogenous (NNARX) networks. The NNARX model is identified from input-output data collected from the plant, and can be given a state-space representation with known measurable states made by past input and output variables, so that a state observer is not required. In the training phase, the Incremental Input-to-State Stability (δISS) property can be forced when consistent with the behavior of the plant. The δISS property is then leveraged to augment the model with an explicit integral action on the output tracking error, which allows to achieve offset-free tracking capabilities to the designed control scheme. The proposed control architecture is numerically tested on a water heating system and the achieved results are compared to those scored by another popular offset-free MPC method, showing that the proposed scheme attains remarkable performances even in presence of disturbances acting on the plant.

Fabio Bonassi, Marcello Farina, Jing Xie et al.

This paper aims to discuss and analyze the potentialities of Recurrent Neural Networks (RNN) in control design applications. The main families of RNN are considered, namely Neural Nonlinear AutoRegressive eXogenous, (NNARX), Echo State Networks (ESN), Long Short Term Memory (LSTM), and Gated Recurrent Units (GRU). The goal is twofold. Firstly, to survey recent results concerning the training of RNN that enjoy Input-to-State Stability (ISS) and Incremental Input-to-State Stability ($δ$ISS) guarantees. Secondly, to discuss the issues that still hinder the widespread use of RNN for control, namely their robustness, verifiability, and interpretability. The former properties are related to the so-called generalization capabilities of the networks, i.e. their consistency with the underlying real plants, even in presence of unseen or perturbed input trajectories. The latter is instead related to the possibility of providing a clear formal connection between the RNN model and the plant. In this context, we illustrate how ISS and $δ$ISS represent a significant step towards the robustness and verifiability of the RNN models, while the requirement of interpretability paves the way to the use of physics-based networks. The design of model predictive controllers with RNN as plant's model is also briefly discussed. Lastly, some of the main topics of the paper are illustrated on a simulated chemical system.

Fabio Bonassi, Riccardo Scattolini

Owing to their superior modeling capabilities, gated Recurrent Neural Networks, such as Gated Recurrent Units (GRUs) and Long Short-Term Memory networks (LSTMs), have become popular tools for learning dynamical systems. This paper aims to discuss how these networks can be adopted for the synthesis of Internal Model Control (IMC) architectures. To this end, first a gated recurrent network is used to learn a model of the unknown input-output stable plant. Then, a controller gated recurrent network is trained to approximate the model inverse. The stability of these networks, ensured by means of a suitable training procedure, allows to guarantee the input-output closed-loop stability. The proposed scheme is able to cope with the saturation of the control variables, and can be deployed on low-power embedded controllers, as it requires limited online computations. The approach is then tested on the Quadruple Tank benchmark system and compared to alternative control laws, resulting in remarkable closed-loop performances.

Fabio Bonassi, C. F. Oliveira da Silva, Riccardo Scattolini

The use of Recurrent Neural Networks (RNNs) for system identification has recently gathered increasing attention, thanks to their black-box modeling capabilities.Albeit RNNs have been fruitfully adopted in many applications, only few works are devoted to provide rigorous theoretical foundations that justify their use for control purposes. The aim of this paper is to describe how stable Gated Recurrent Units (GRUs), a particular RNN architecture, can be trained and employed in a Nonlinear MPC framework to perform offset-free tracking of constant references with guaranteed closed-loop stability. The proposed approach is tested on a pH neutralization process benchmark, showing remarkable performances.

Fabio Bonassi, Marcello Farina, Riccardo Scattolini

The goal of this paper is to provide sufficient conditions for guaranteeing the Input-to-State Stability (ISS) and the Incremental Input-to-State Stability (δISS) of Gated Recurrent Units (GRUs) neural networks. These conditions, devised for both single-layer and multi-layer architectures, consist of nonlinear inequalities on network's weights. They can be employed to check the stability of trained networks, or can be enforced as constraints during the training procedure of a GRU. The resulting training procedure is tested on a Quadruple Tank nonlinear benchmark system, showing satisfactory modeling performances.

Simone Pozzoli, Marco Gallieri, Riccardo Scattolini

The use of recurrent neural networks to represent the dynamics of unstable systems is difficult due to the need to properly initialize their internal states, which in most of the cases do not have any physical meaning, consequent to the non-smoothness of the optimization problem. For this reason, in this paper focus is placed on mechanical systems characterized by a number of degrees of freedom, each one represented by two states, namely position and velocity. For these systems, a new recurrent neural network is proposed: Tustin-Net. Inspired by second-order dynamics, the network hidden states can be straightforwardly estimated, as their differential relationships with the measured states are hardcoded in the forward pass. The proposed structure is used to model a double inverted pendulum and for model-based Reinforcement Learning, where an adaptive Model Predictive Controller scheme using the Unscented Kalman Filter is proposed to deal with parameter changes in the system.

Marcello Farina, Xinglong Zhang, Riccardo Scattolini

This paper presents a hierarchical control scheme for interconnected linear systems. At the higher layer of the control structure a robust centralized Model Predictive Control (MPC) algorithm based on a reduced order dynamic model of the overall system optimizes a long-term performance index penalizing the deviation of the state and the control input from their nominal values. At the lower layer local MPC regulators, possibly working at different rates, are designed for the full order models of the subsystems to refine the control action computed at the higher layer. A simulation experiment is presented to describe the implementation aspects and the potentialities of the proposed approach.