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In this paper we propose novel global and regional stability analysis conditions based on linear matrix inequalities for a general class of recurrent neural networks. These conditions can be also used for state-feedback control design and a suitable optimization problem enforcing H2 norm minimization properties is defined. The theoretical results are corroborated by numerical simulations, showing the advantages and limitations of the methods presented herein.

This report presents three Moving Horizon Estimation (MHE) methods for discrete-time partitioned linear systems, i.e. systems decomposed into coupled subsystems with non-overlapping states. The MHE approach is used due to its capability of exploiting physical constraints on states in the estimation process. In the proposed algorithms, each subsystem solves reduced-order MHE problems to estimate its own state and different estimators have different computational complexity, accuracy and transmission requirements among subsystems. In all cases, conditions for the convergence of the estimation error to zero are analyzed.

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., Comment: {\copyright} 2023. This manuscript version is made available under the CC-BY-NC-ND 4.0 license (https://creativecommons.org/licenses/by-nc-nd/4.0/). This manuscript has been accepted for publication at Elsevier Automatica. Please cite the published article instead of this manuscript

Optimal management of thermal and energy grids with fluctuating demand and prices requires to orchestrate the generation units (GU) among all their operating modes. A hierarchical approach is proposed to control coupled energy nonlinear systems. The high level hybrid optimization defines the unit commitment, with the optimal transition strategy, and best production profiles. The low level dynamic model predictive control (MPC), receiving the set-points from the upper layer, safely governs the systems considering process constraints. To enhance the overall efficiency of the system, a method to optimal start-up the GU is here presented: a linear parameter varying MPC computes the optimal trajectory in closed-loop by iteratively linearising the system along the previous optimal solution. The introduction of an intermediate equilibrium state as additional decision variable permits the reduction of the optimization horizon,while a terminal cost term steers the system to the target set-point. Simulation results show the effectiveness of the proposed approach.

A hierarchical architecture for the optimal management of an ensemble of steam generators is presented. The subsystems are coordinated by a multilayer scheme for jointly sustaining a common load. The high level optimizes the load allocation and the generator schedule, considering activation dynamics by a hybrid model. At the medium level, a robust tube-based model predictive control (MPC) tracks a time-varying demand using a centralized--but aggregate--model, whose order does not scale with the number of subsystems. A nonlinear optimization, at medium level, addresses MPC infeasibility due to abrupt changes of ensemble configuration. Low-level decentralized controllers stabilize the generators. This control scheme enables the dynamical modification of the ensemble configuration and plug and play operations. Simulations demonstrate the approach potentialities.

The aim of this work is to investigate the use of Incrementally Input-to-State Stable ($\delta$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-$\delta$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., Comment: This manuscript is an extended version of a paper accepted for the 2023 European Control Conference (ECC'23). Copyright may be transferred without notice, after which this version may no longer be accessible

This paper proposes a novel sufficient condition for the incremental input-to-state stability of a generic 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.

This paper presents a robust Model Predictive Control (MPC) scheme that provides offset-free setpoint tracking for systems described by Neural Nonlinear AutoRegressive eXogenous (NNARX) models. The NNARX model learns the dynamics of the plant from input-output data, and during the training the Incremental Input-to-State Stability (${\delta}$ISS) property is forced to guarantee stability. The trained NNARX model is then augmented with an explicit integral action on the output tracking error, which allows the control scheme to enjoy offset-free tracking ability. A tube-based MPC is finally designed, leveraging the unique structure of the model, to ensure robust stability and robust asymptotic zero error regulation for constant reference signals in the presence of model-plant mismatch or unknown disturbances. Numerical simulations on a water heating system show the effectiveness of the proposed control algorithm., Comment: This is the peer reviewed version of http://doi.org/10.1002/rnc.6883, available in Open Access on the publisher's website. Please, cite Reference 16 instead

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., Comment: Copyright 2022 EUCA. This article appears in the Proceedings of the 2022 European Control Conference (ECC'22), July 12-15, 2022, London, pp. 2018-2023

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 ({\delta}ISS) property can be forced when consistent with the behavior of the plant. The {\delta}ISS 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., Comment: {\copyright} 2022 IEEE. This manuscript is the extended version of an article appearing in the Proceedings of the 2022 IEEE 61st Conference on Decision and Control (CDC), July 12-15, 2022, Cancun, Mexico, pp. 2123-2128. DOI: 10.1109/CDC51059.2022.9992362

In this paper we propose a novel methodology that allows to design, in a purely data-based fashion and for linear single-input and single-output systems, both robustly stable and performing control systems for tracking piecewise constant reference signals. The approach uses both (i) Virtual Reference Feedback Tuning for enforcing suitable performances and (ii) the Set Membership framework for providing a-priori robust stability guarantees. Indeed, an uncertainty set for the system parameters is obtained through Set Membership identification, where an algorithm based on the scenario approach is proposed to estimate the inflation parameter in a probabilistic way. Based on this set, robust stability conditions are enforced as Linear Matrix Inequality constraints within an optimization problem whose linear cost function relies on Virtual Reference Feedback Tuning. To show the generality and effectiveness of our approach, we apply it to two of the most widely used yet simple control schemes, i.e., where tracking is achieved thanks to (i) a static feedforward action and (ii) an integrator in closed-loop. The proposed method is not fully direct due to the Set Membership identification. However, the uncertainty set is used with the only objective of providing robust stability guarantees for the closed-loop system and it is not directly used for the performances optimization, which instead is totally based on data. The effectiveness of the developed method is demonstrated with reference to two simulation examples. A comparison with other data-driven methods is also carried out.

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 ($\delta$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 $\delta$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., Comment: Copyright 2022. This manuscript version is made available under the CC-BY-NC-ND 4.0 license. This manuscript has been published at Elsevier Journal of Process Control. Published article available at DOI https://doi.org/10.1016/j.jprocont.2022.04.011

In this paper the application of Virtual Reference Feedback Tuning (VRFT) for control of nonlinear systems with regulators defined by Echo State Networks (ESN) and Long Short Term Memory (LSTM) networks is investigated. The capability of this class of regulators of constraining the control variable is pointed out and an advanced control scheme that allows to achieve zero steady-state error is presented. The developed algorithms are validated on a benchmark example that consists of an electronic throttle body (ETB).

This note extends a recently proposed algorithm for model identification and robust MPC of asymptotically stable, linear time-invariant systems subject to process and measurement disturbances. Independent output predictors for different steps ahead are estimated with Set Membership methods. It is here shown that the corresponding prediction error bounds are the least conservative in the considered model class. Then, a new multi-rate robust MPC algorithm is developed, employing said multi-step predictors to robustly enforce constraints and stability against disturbances and model uncertainty, and to reduce conservativeness. A simulation example illustrates the effectiveness of the approach.

The idea of using Feed-Forward Neural Networks (FFNNs) as regression functions for Nonlinear AutoRegressive eXogenous (NARX) models, leading to models herein named Neural NARXs (NNARXs), has been quite popular in the early days of machine learning applied to nonlinear system identification, owing to their simple structure and ease of application to control design. Nonetheless, few theoretical results are available concerning the stability properties of these models. In this paper we address this problem, providing a sufficient condition under which NNARX models are guaranteed to enjoy the Input-to-State Stability (ISS) and the Incremental Input-to-State Stability ({\delta}ISS) properties. This condition, which is an inequality on the weights of the underlying FFNN, can be enforced during the training procedure to ensure the stability of the model. The proposed model, along with this stability condition, are tested on the pH neutralization process benchmark, showing satisfactory results., Comment: Copyright 2021 by the authors. This work has been accepted to IFAC (19th IFAC Symposium on System Identification: learning models for decision and control) for publication under a Creative Commons Licence CC-BY-NC-ND. Published article: https://doi.org/10.1016/j.ifacol.2021.08.417

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 ({\delta}ISS) 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., Comment: Copyright 2021. This manuscript version is made available under the CC-BY-NC-ND 4.0

This work presents a hierarchical architecture for the optimal management of an ensemble of steam generators, which needs to jointly sustain a common load. The coordination of independent subsystems is provided by a multi-layer control scheme. A high-level optimizer computes the optimal shares of production to be allocated to single generators. At medium level, a robust tube-based model predictive control (MPC) is proposed to track the time-varying demand of the ensemble using a centralized, but aggregated model, whose order does not scale with the number of subsystems. At low level, decentralized controllers are in place to stabilize the internal boiler pressure. The control architecture enables the dynamic modification of the ensemble configuration and plug and play operations. Simulation results are reported to demonstrate the potentialities of the proposed approach.

This paper proposes a feedback linearizing law for single-track dynamic models, allowing the design of a trajectory tracking controller exploiting linear control theory. The main characteristics of this algorithm are its simplicity, its independence from any vehicle model parameter, apart from the position of the center of mass, and its robustness. In particular, a numerical bifurcation analysis demonstrates that, for physically meaningful values of the center of mass deviation, the equilibrium is structurally asymptotically stable. Experimental results, concerning the linearising law and its application as inner loop of a trajectory tracking controller, are also presented, confirming the effectiveness of the proposal.

The goal of this paper is to analyze Long Short Term Memory (LSTM) neural networks from a dynamical system perspective. The classical recursive equations describing the evolution of LSTM can be recast in state space form, resulting in a time-invariant nonlinear dynamical system. A sufficient condition guaranteeing the Input-to-State (ISS) stability property of this class of systems is provided. The ISS property entails the boundedness of the output reachable set of the LSTM. In light of this result, a novel approach for the safety verification of the network, based on the Scenario Approach, is devised. The proposed method is eventually tested on a pH neutralization process., Comment: Accepted for Learning for dynamics & control (L4DC) 2020

Echo State Networks (ESNs) are widely-used Recurrent Neural Networks. They are dynamical systems including, in state-space form, a nonlinear state equation and a linear output transformation. The common procedure to train ESNs is to randomly select the parameters of the state equation, and then to estimate those of the output equation via a standard least squares problem. Such a procedure is repeated for different instances of the random parameters characterizing the state equation, until satisfactory results are achieved. However, this trial-and-error procedure is not systematic and does not provide any guarantee about the optimality of the identification results. To solve this problem, we propose to complement the identification procedure of ESNs by applying results in scenario optimization. The resulting training procedure is theoretically sound and allows one to link precisely the number of identification instances to a guaranteed optimality bound on relevant performance indexes, such as the Root Mean Square error and the FIT index of the estimated model evaluated over a validation data-set. The proposed procedure is finally applied to the simulated model of a pH neutralization process: the obtained results confirm the validity of the approach., Comment: 6 pages

In this paper, we propose, discuss, and validate an online Nonlinear Model Predictive Control (NMPC) method for multi-rotor aerial systems with arbitrarily positioned and oriented rotors which simultaneously addresses the local reference trajectory planning and tracking problems. This work brings into question some common modeling and control design choices that are typically adopted to guarantee robustness and reliability but which may severely limit the attainable performance. Unlike most of state of the art works, the proposed method takes advantages of a unified nonlinear model which aims to describe the whole robot dynamics by explicitly including a realistic physical description of the actuator dynamics and limitations. As a matter of fact, our solution does not resort to common simplifications such as: 1) linear model approximation, 2) cascaded control paradigm used to decouple the translational and the rotational dynamics of the rigid body, 3) use of low-level reactive trackers for the stabilization of the internal loop, and 4) unconstrained optimization resolution or use of fictitious constraints. More in detail, we consider as control inputs the derivatives of the propeller forces and propose a novel method to suitably identify the actuator limitations by leveraging experimental data. Differently from previous approaches, the constraints of the optimization problem are defined only by the real physics of the actuators, avoiding conservative -- and often not physical -- input/state saturations which are present, e.g., in cascaded approaches. The control algorithm is implemented using a state-of-the-art Real Time Iteration (RTI) scheme with partial sensitivity update method. CONTINUES..., Comment: cs.RO cs.SY math.OC math.DS

This paper analyzes the stability-related properties of Long Short-Term Memory (LSTM) networks and investigates their use as the model of the plant in the design of Model Predictive Controllers (MPC). First, sufficient conditions guaranteeing the Input-to-State stability (ISS) and Incremental Input-to-State stability (dISS) of LSTM are derived. These properties are then exploited to design an observer with guaranteed convergence of the state estimate to the true one. Such observer is then embedded in a MPC scheme solving the tracking problem. The resulting closed-loop scheme is proved to be asymptotically stable. The training algorithm and control scheme are tested numerically on the simulator of a pH reactor, and the reported results confirm the effectiveness of the proposed approach.

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., Comment: 6 pages,5 figures, submitted to European Control Conference (ECC)

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.

The increasing diffusion of distributed energy generation systems requires the development of new control paradigms for the coordination of micro-generators, storage systems, and loads aimed at maintaining the efficiency and the safe operability of the electricity network. MicroGrids (MGs) are an interesting way to locally manage groups of generation devices, but they cannot singularly provide a significant contribution to sustain the main electricity grid in terms of ancillary services, such as the availability of a minimum amount of power reserve for the frequency regulation. For these reasons, in this paper we propose a framework for the aggregation and coordination of interconnected MGs to provide ancillary services to the main utility. The proposed framework is structured in three main phases. In the first one, a distributed optimization algorithm computes the day-ahead profile of the active power production of the MGs based on the available forecasts of the renewable sources production and the loads absorption. In this phase, scalability of the optimization problem and confidentiality requirements are guaranteed. In the second phase, reactive power flows are scheduled and it is ensured that the active power trends planned in the first phase do not compromise the voltage/current limitations. A final third phase is used to schedule the active and reactive power profiles of the generation units of each MG to make them consistent with the requirements and results of the previous two phases. The developed method is used for control of the IEEE 13-bus system network and the results achieved are thoroughly discussed in terms of performance and scalability properties., Comment: (under revision)

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., Comment: This manuscript contains technical details of recent results developed by the authors on learning-based model predictive control for linear time invariant systems

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., Comment: 35 pages,7 figures

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.

In this paper we propose a novel partition-based distributed state estimation scheme for non-overlapping subsystems based on Kalman filter. The estimation scheme is designed in order to account, in a rigorous fashion, for dynamic coupling terms between subsystems, and for the uncertainty related to the state estimates performed by the neighboring subsystems. The online implementation of the proposed estimation scheme is scalable, since it involves (i) small-scale matrix operations to be carried out by the estimator embedded in each subsystem and (ii) neighbor-to-neighbor transmission of a limited amount of data. We provide theoretical conditions ensuring the estimation convergence. Reconfigurability of the proposed estimation scheme is allowed in case of plug and play operations. Simulation tests are provided to illustrate the effectiveness of the proposed algorithm.

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.

The Model Predictive Control (MPC) approach is used in this paper to control the voltage profiles in MV networks with distributed generation. The proposed algorithm lies at the intermediate level of a three-layer hierarchical structure. At the upper level a static Optimal Power Flow (OPF) manager computes the required voltage profiles to be transmitted to the MPC level, while at the lower level local Automatic Voltage Regulators (AVR), one for each Distributed Generator (DG), track the reactive power reference values computed by MPC. The control algorithm is based on an impulse response model of the system, easily obtained by means of a detailed simulator of the network, and allows to cope with constraints on the voltage profiles and/or on the reactive power flows along the network. If these constraints cannot be satisfied by acting on the available DGs, the algorithm acts on the On-Load Tap Changing (OLTC) transformer. A radial rural network with two feeders, eight DGs, and thirty-one loads is used as case study. The model of the network is implemented in DIgSILENT PowerFactory, while the control algorithm runs in Matlab. A number of simulation results is reported to witness the main characteristics and limitations of the proposed approach.

This paper proposes a state estimator for large-scale linear systems described by the interaction of state-coupled subsystems affected by bounded disturbances. We equip each subsystem with a Local State Estimator (LSE) for the reconstruction of the subsystem states using pieces of information from parent subsystems only. Moreover we provide conditions guaranteeing that the estimation errors are confined into prescribed polyhedral sets and converge to zero in absence of disturbances. Quite remarkably, the design of an LSE is recast into an optimization problem that requires data from the corresponding subsystem and its parents only. This allows one to synthesize LSEs in a Plug-and-Play (PnP) fashion, i.e. when a subsystem gets added, the update of the whole estimator requires at most the design of an LSE for the subsystem and its parents. Theoretical results are backed up by numerical experiments on a mechanical system.