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In this paper, we present an early software integrating Reinforcement Learning (RL) with Model Predictive Control (MPC). Our aim is to make recent theoretical contributions from the literature more accessible to both the RL and MPC communities. We combine standard software tools developed by the RL community, such as Gymnasium, stable-baselines3, or CleanRL with the acados toolbox, a widely-used software package for efficient MPC algorithms. Our core contribution is MPC4RL, an open-source Python package that supports learning-enhanced MPC schemes for existing acados implementations. The package is designed to be modular, extensible, and user-friendly, facilitating the tuning of MPC algorithms for a broad range of control problems. It is available on GitHub.

AI models that predict the future behavior of a system (a.k.a. predictive AI models) are central to intelligent decision-making. However, decision-making using predictive AI models often results in suboptimal performance. This is primarily because AI models are typically constructed to best fit the data, and hence to predict the most likely future rather than to enable high-performance decision-making. The hope that such prediction enables high-performance decisions is neither guaranteed in theory nor established in practice. In fact, there is increasing empirical evidence that predictive models must be tailored to decision-making objectives for performance. In this paper, we establish formal (necessary and sufficient) conditions that a predictive model (AI-based or not) must satisfy for a decision-making policy established using that model to be optimal. We then discuss their implications for building predictive AI models for sequential decision-making.

Optimality is a critical aspect of Model Predictive Control (MPC), especially in economic MPC. However, achieving optimality in MPC presents significant challenges, and may even be impossible, due to inherent inaccuracies in the predictive models. Predictive models often fail to accurately capture the true system dynamics, such as in the presence of stochasticity, leading to suboptimal MPC policies. In this paper, we establish the necessary and sufficient conditions on the underlying prediction model for an MPC scheme to achieve closed-loop optimality. Interestingly, these conditions are counterintuitive to the traditional approach of building predictive models that best fit the data. These conditions present a mathematical foundation for constructing models that are directly linked to the performance of the resulting MPC scheme., Comment: 7 pages, submitted to Automatica as a brief paper

Wastewater treatment plants face unique challenges for process control due to their complex dynamics, slow time constants, and stochastic delays in observations and actions. These characteristics make conventional control methods, such as Proportional-Integral-Derivative controllers, suboptimal for achieving efficient phosphorus removal, a critical component of wastewater treatment to ensure environmental sustainability. This study addresses these challenges using a novel deep reinforcement learning approach based on the Soft Actor-Critic algorithm, integrated with a custom simulator designed to model the delayed feedback inherent in wastewater treatment plants. The simulator incorporates Long Short-Term Memory networks for accurate multi-step state predictions, enabling realistic training scenarios. To account for the stochastic nature of delays, agents were trained under three delay scenarios: no delay, constant delay, and random delay. The results demonstrate that incorporating random delays into the reinforcement learning framework significantly improves phosphorus removal efficiency while reducing operational costs. Specifically, the delay-aware agent achieved 36% reduction in phosphorus emissions, 55% higher reward, 77% lower target deviation from the regulatory limit, and 9% lower total costs than traditional control methods in the simulated environment. These findings underscore the potential of reinforcement learning to overcome the limitations of conventional control strategies in wastewater treatment, providing an adaptive and cost-effective solution for phosphorus removal.

Safe reinforcement learning (RL) is a promising approach for many real-world decision-making problems where ensuring safety is a critical necessity. In safe RL research, while expected cumulative safety constraints (ECSCs) are typically the first choices, chance constraints are often more pragmatic for incorporating safety under uncertainties. This paper proposes a \textit{flipping-based policy} for Chance-Constrained Markov Decision Processes (CCMDPs). The flipping-based policy selects the next action by tossing a potentially distorted coin between two action candidates. The probability of the flip and the two action candidates vary depending on the state. We establish a Bellman equation for CCMDPs and further prove the existence of a flipping-based policy within the optimal solution sets. Since solving the problem with joint chance constraints is challenging in practice, we then prove that joint chance constraints can be approximated into Expected Cumulative Safety Constraints (ECSCs) and that there exists a flipping-based policy in the optimal solution sets for constrained MDPs with ECSCs. As a specific instance of practical implementations, we present a framework for adapting constrained policy optimization to train a flipping-based policy. This framework can be applied to other safe RL algorithms. We demonstrate that the flipping-based policy can improve the performance of the existing safe RL algorithms under the same limits of safety constraints on Safety Gym benchmarks., Comment: Accepted to NeurIPS 2024

This paper proposes a model predictive controller for discrete-time linear systems with additive, possibly unbounded, stochastic disturbances and subject to chance constraints. By computing a polytopic probabilistic positively invariant set for constraint tightening with the help of the computation of the minimal robust positively invariant set, the chance constraints are guaranteed, assuming only the mean and covariance of the disturbance distribution are given. The resulting online optimization problem is a standard strictly quadratic programming, just like in conventional model predictive control with recursive feasibility and stability guarantees and is simple to implement. A numerical example is provided to illustrate the proposed method., Comment: This paper was accepted for publication in CDC 2024

Markov Decision Processes (MDPs) offer a fairly generic and powerful framework to discuss the notion of optimal policies for dynamic systems, in particular when the dynamics are stochastic. However, computing the optimal policy of an MDP can be very difficult due to the curse of dimensionality present in solving the underlying Bellman equations. Model Predictive Control (MPC) is a very popular technique for building control policies for complex dynamic systems. Historically, MPC has focused on constraint satisfaction and steering dynamic systems towards a user-defined reference. More recently, Economic MPC was proposed as a computationally tractable way of building optimal policies for dynamic systems. When stochsaticity is present, economic MPC is close to the MDP framework. In that context, Economic MPC can be construed as attractable heuristic to provide approximate solutions to MDPs. However, there is arguably a knowledge gap in the literature regarding these approximate solutions and the conditions for an MPC scheme to achieve closed-loop optimality. This chapter aims to clarify this approximation pedagogically, to provide the conditions for MPC to deliver optimal policies, and to explore some of their consequences.

In this chapter, we report on our experience with domestic flexible electric energy demand based on a regular commercial (HVAC)-based heating system in a house. Our focus is on investigating the predictability of the energy demand of the heating system and of the thermal response when varying the heating system settings. Being able to form such predictions is crucial for most flexible demand algorithms. We will compare several methods for predicting the thermal and energy response, which either gave good results or which are currently promoted in the literature for controlling buildings. We will report that the stochasticity of a house response is -- in our experience -- the main difficulty in providing domestic flexible demand from heating. The experiments were carried out on a regular house in Norway, equipped with four air-to-air Mitsubishi heat pumps and a high-efficiency balanced ventilation system. The house was equipped with multiple IoT-based climate sensors, real-time power measurement, and the possibility to drive the HVAC system via the IoT. The house is operating on the spot market (Nord Pool NO3) and is exposed to a peak energy demand penalty. Over a period of three years, we have collected data on the house (temperatures, humidity, air quality), real-time power and hourly energy consumption, while applying various flexible demand algorithms responding to the local energy costs. This has produced large variations in the settings of the heating system and energy demand, resulting in rich data for investigating the house response. This chapter aims at providing important insights on providing flexible demand from houses in cold climates.

In this paper, we explore the interplay between Predictive Control and closed-loop optimality, spanning from Model Predictive Control to Data-Driven Predictive Control. Predictive Control in general relies on some form of prediction scheme on the real system trajectories. However, these predictions may not accurately capture the real system dynamics, for e.g., due to stochasticity, resulting in sub-optimal control policies. This lack of optimality is a critical issue in case of problems with economic objectives. We address this by providing sufficient conditions on the underlying prediction scheme such that a Predictive Controller can achieve closed-loop optimality. However, these conditions do not readily extend to Data-Driven Predictive Control. In this context of closed-loop optimality, we conclude that the factor distinguishing the approaches within Data-Driven Predictive Control is if they can be cast as a sequential decision-making process or not, rather than the dichotomy of model-based vs. model-free. Furthermore, we show that the conventional approach of improving the prediction accuracy from data may not guarantee optimality.

Validating and controlling safety-critical systems in uncertain environments necessitates probabilistic reachable sets of future state evolutions. The existing methods of computing probabilistic reachable sets normally assume that stochastic uncertainties are independent of system states, inputs, and other environment variables. However, this assumption falls short in many real-world applications, where the probability distribution governing uncertainties depends on these variables, referred to as contextual uncertainties. This paper addresses the challenge of computing probabilistic reachable sets of stochastic nonlinear states with contextual uncertainties by seeking minimum-volume polynomial sublevel sets with contextual chance constraints. The formulated problem cannot be solved by the existing sample-based approximation method since the existing methods do not consider conditional probability densities. To address this, we propose a consistent sample approximation of the original problem by leveraging conditional density estimation and resampling. The obtained approximate problem is a tractable optimization problem. Additionally, we prove the proposed sample-based approximation's almost uniform convergence, showing that it gives the optimal solution almost consistently with the original ones. Through a numerical example, we evaluate the effectiveness of the proposed method against existing approaches, highlighting its capability to significantly reduce the bias inherent in sample-based approximation without considering a conditional probability density.

With the increasing number of fast-electric vehicle charging stations (fast-EVCSs) and the popularization of information technology, electricity price competition between fast-EVCSs is highly expected, in which the utilization of public and/or privacy-preserved information will play a crucial role. Self-interest electric vehicle (EV) users, on the other hand, try to select a fast-EVCS for charging in a way to maximize their utilities based on electricity price, estimated waiting time, and their state of charge. While existing studies have largely focused on finding equilibrium prices, this study proposes a personalized dynamic pricing policy (PeDP) for a fast-EVCS to maximize revenue using a reinforcement learning (RL) approach. We first propose a multiple fast-EVCSs competing simulation environment to model the selfish behavior of EV users using a game-based charging station selection model with a monetary utility function. In the environment, we propose a Q-learning-based PeDP to maximize fast-EVCS' revenue. Through numerical simulations based on the environment: (1) we identify the importance of waiting time in the EV charging market by comparing the classic Bertrand competition model with the proposed PeDP for fast-EVCSs (from the system perspective); (2) we evaluate the performance of the proposed PeDP and analyze the effects of the information on the policy (from the service provider perspective); and (3) it can be seen that privacy-preserved information sharing can be misused by artificial intelligence-based PeDP in a certain situation in the EV charging market (from the customer perspective).

The last few years have seen the massive deployment of electric buses in many existing transit networks. However, the planning and operation of an electric bus system differ from that of a bus system with conventional vehicles, and some key problems have not yet been studied in the literature. In this work, we address the integrated operational control and charging scheduling problem for a network of electric buses with a limited opportunity charging capacity. We propose a hierarchical control framework to solve this problem, where the charging and operational decisions are taken jointly by solving a mixed-integer linear program in the high-level control layer. Since this optimization problem might become very large as more bus lines are considered, we propose to apply Lagrangian relaxation in such a way as to exploit the structure of the problem and enable a decomposition into independent subproblems. A local search heuristic is then deployed in order to generate good feasible solutions to the original problem. This entire Lagrangian heuristic procedure is shown to scale much better on transit networks with an increasing number of bus lines than trying to solve the original problem with an off-the-shelf solver. The proposed procedure is then tested in the high-fidelity microscopic traffic environment Vissim on a bus network constructed from an openly available dataset of the city of Chicago. The results show the benefits of combining the charging scheduling decisions together with the real-time operational control of the vehicles as the proposed control framework manages to achieve both a better level of service and lower charging costs over control baselines with predetermined charging schedules., Comment: 29 pages, 9 figures

A novel perspective on the design of robust model predictive control (MPC) methods is presented, whereby closed-loop constraint satisfaction is ensured using recursive feasibility of the MPC optimization. Necessary and sufficient conditions are derived for recursive feasibility, based on the effects of model perturbations and disturbances occurring at one time step. Using these conditions and Farkas' lemma, sufficient conditions suitable for design are formulated. The proposed method is called a closed-loop design, as only the existence of feasible inputs at the next time step is enforced by design. This is in contrast to most existing formulations, which compute control policies that are feasible under the worst-case realizations of all model perturbations and exogenous disturbances in the MPC prediction horizon. The proposed method has an online computational complexity similar to nominal MPC methods while preserving guarantees of constraint satisfaction, recursive feasibility and stability. Numerical simulations demonstrate the efficacy of our proposed approach., Comment: Submitted to IEEE Transactions on Automatic Control

The exploding research interest for neural networks in modeling nonlinear dynamical systems is largely explained by the networks' capacity to model complex input-output relations directly from data. However, they typically need vast training data before they can be put to any good use. The data generation process for dynamical systems can be an expensive endeavor both in terms of time and resources. Active learning addresses this shortcoming by acquiring the most informative data, thereby reducing the need to collect enormous datasets. What makes the current work unique is integrating the deep active learning framework into nonlinear system identification. We formulate a general static deep active learning acquisition problem for nonlinear system identification. This is enabled by exploring system dynamics locally in different regions of the input space to obtain a simulated dataset covering the broader input space. This simulated dataset can be used in a static deep active learning acquisition scheme referred to as global explorations. The global exploration acquires a batch of initial states corresponding to the most informative state-action trajectories according to a batch acquisition function. The local exploration solves an optimal control problem, finding the control trajectory that maximizes some measure of information. After a batch of informative initial states is acquired, a new round of local explorations from the initial states in the batch is conducted to obtain a set of corresponding control trajectories that are to be applied on the system dynamics to get data from the system. Information measures used in the acquisition scheme are derived from the predictive variance of an ensemble of neural networks. The novel method outperforms standard data acquisition methods used for system identification of nonlinear dynamical systems in the case study performed on simulated data.

This paper presents an approach for learning Model Predictive Control (MPC) schemes directly from data using Reinforcement Learning (RL) methods. The state-of-the-art learning methods use RL to improve the performance of parameterized MPC schemes. However, these learning algorithms are often gradient-based methods that require frequent evaluations of computationally expensive MPC schemes, thereby restricting their use on big datasets. We propose to tackle this issue by using tools from RL to learn a parameterized MPC scheme directly from data in an offline fashion. Our approach derives an MPC scheme without having to solve it over the collected dataset, thereby eliminating the computational complexity of existing techniques for big data. We evaluate the proposed method on three simulated experiments of varying complexity.

The rapid adoption of electric buses by transit agencies around the world is leading to new challenges in the planning and operation of bus networks. In particular, the limited driving range of electric vehicles imposes operational constraints such as the need to charge buses during service. Research on this topic has mostly focused on the strategic and tactical planning aspects until now, and very little work has been done on the real-time operational aspect. To remedy this, we propose integrating the charging scheduling problem with a real-time speed control strategy in this paper. The control problem is formulated as a mixed-integer linear program and solved to optimality with the branch-and-bound method. Simulations are carried out by repeatedly solving the control problem in a receding horizon fashion over a full day of operation. The results show that the proposed controller manages to anticipate and avoid potential conflicts where the charging demand exceeds the charger capacity. This feature makes the controller achieve lower operational costs, both in terms of service regularity and energy consumption, compared to a standard first-come, first-served charging scheme., Comment: 6 pages, 3 figures

This paper shows that the optimal policy and value functions of a Markov Decision Process (MDP), either discounted or not, can be captured by a finite-horizon undiscounted Optimal Control Problem (OCP), even if based on an inexact model. This can be achieved by selecting a proper stage cost and terminal cost for the OCP. A very useful particular case of OCP is a Model Predictive Control (MPC) scheme where a deterministic (possibly nonlinear) model is used to reduce the computational complexity. This observation leads us to parameterize an MPC scheme fully, including the cost function. In practice, Reinforcement Learning algorithms can then be used to tune the parameterized MPC scheme. We verify the developed theorems analytically in an LQR case and we investigate some other nonlinear examples in simulations., Comment: 8 pages and 10 figures

This paper aims to provide a Dynamic Programming (DP) approach to solve the Mission-Wide Chance-Constrained Optimal Control Problems (MWCC-OCP). The mission-wide chance constraint guarantees that the probability that the entire state trajectory lies within a constraint/safe region is higher than a prescribed level, and is different from the stage-wise chance constraints imposed at individual time steps. The control objective is to find an optimal policy sequence that achieves both (i) satisfaction of a mission-wide chance constraint, and (ii) minimization of a cost function. By transforming the stage-wise chance-constrained problem into an unconstrained counterpart via Lagrangian method, standard DP can then be deployed. Yet, for MWCC-OCP, this methods fails to apply, because the mission-wide chance constraint cannot be easily formulated using stage-wise chance constraints due to the time-correlation between the latter (individual states are coupled through the system dynamics). To fill this gap, firstly, we detail the conditions required for a classical DP solution to exist for this type of problem; secondly, we propose a DP solution to the MWCC-OCP through state augmentation by introducing an additional functional state variable.

In this paper, we propose a learning-based Model Predictive Control (MPC) approach for the polytopic Linear Parameter-Varying (LPV) systems with inexact scheduling parameters (as exogenous signals with inexact bounds), where the Linear Time Invariant (LTI) models (vertices) captured by combinations of the scheduling parameters becomes wrong. We first propose to adopt a Moving Horizon Estimation (MHE) scheme to simultaneously estimate the convex combination vector and unmeasured states based on the observations and model matching error. To tackle the wrong LTI models used in both the MPC and MHE schemes, we then adopt a Policy Gradient (PG) Reinforcement Learning (RL) to learn both the estimator (MHE) and controller (MPC) so that the best closed-loop performance is achieved. The effectiveness of the proposed RL-based MHE/MPC design is demonstrated using an illustrative example.

Reinforcement learning methods typically use Deep Neural Networks to approximate the value functions and policies underlying a Markov Decision Process. Unfortunately, DNN-based RL suffers from a lack of explainability of the resulting policy. In this paper, we instead approximate the policy and value functions using an optimization problem, taking the form of Quadratic Programs (QPs). We propose simple tools to promote structures in the QP, pushing it to resemble a linear MPC scheme. A generic unstructured QP offers high flexibility for learning, while a QP having the structure of an MPC scheme promotes the explainability of the resulting policy, additionally provides ways for its analysis. The tools we propose allow for continuously adjusting the trade-off between the former and the latter during learning. We illustrate the workings of our proposed method with the resulting structure using a point-mass task.

Battery cycle life prediction using early degradation data has many potential applications throughout the battery product life cycle. For that reason, various data-driven methods have been proposed for point prediction of battery cycle life with minimum knowledge of the battery degradation mechanisms. However, managing the rapidly increasing amounts of batteries at end-of-life with lower economic and technical risk requires prediction of cycle life with quantified uncertainty, which is still lacking. The interpretability (i.e., the reason for high prediction accuracy) of these advanced data-driven methods is also worthy of investigation. Here, a Quantile Regression Forest (QRF) model, having the advantage of not assuming any specific distribution of cycle life, is introduced to make cycle life range prediction with uncertainty quantified as the width of the prediction interval, in addition to point predictions with high accuracy. The hyperparameters of the QRF model are optimized with a proposed alpha-logistic-weighted criterion so that the coverage probabilities associated with the prediction intervals are calibrated. The interpretability of the final QRF model is explored with two global model-agnostic methods, namely permutation importance and partial dependence plot.

This paper discusses the functional stability of closed-loop Markov Chains under optimal policies resulting from a discounted optimality criterion, forming Markov Decision Processes (MDPs). We investigate the stability of MDPs in the sense of probability measures (densities) underlying the state distributions and extend the dissipativity theory of Economic Model Predictive Control in order to characterize the MDP stability. This theory requires a so-called storage function satisfying a dissipativity inequality. In the probability measures space and for the discounted setting, we introduce new dissipativity conditions ensuring the MDP stability. We then use finite-horizon optimal control problems in order to generate valid storage functionals. In practice, we propose to use Q-learning to compute the storage functionals., Comment: This paper has been accepted to ECC2022. 6 pages

This paper presents a model-free approximation for the Hessian of the performance of deterministic policies to use in the context of Reinforcement Learning based on Quasi-Newton steps in the policy parameters. We show that the approximate Hessian converges to the exact Hessian at the optimal policy, and allows for a superlinear convergence in the learning, provided that the policy parametrization is rich. The natural policy gradient method can be interpreted as a particular case of the proposed method. We analytically verify the formulation in a simple linear case and compare the convergence of the proposed method with the natural policy gradient in a nonlinear example., Comment: This paper has been accepted to 2022 American Control Conference (ACC). 6 pages

The aim of this paper is to propose a high performance control approach for trajectory tracking of Autonomous Underwater Vehicles (AUVs). However, the controller performance can be affected by the unknown perturbations including model uncertainties and external time-varying disturbances in an undersea environment. To address this problem, a Backstepping-based Integral Sliding Mode Control with Time Delay Estimation (BS-ISMC-TDE) is designed. To improve the performance of a conventional backstepping control algorithm, an Integral Sliding Mode Control (ISMC) approach is adopted in the backstepping design to attenuate the steady-state error. Moreover, an adaptive Time Delay Estimation (TDE) strategy is proposed to provide an estimation of perturbations by observing the inputs and the states of the AUV one step into the past without an exact knowledge of the dynamics and the upper bound of uncertainties. From the simulation results, it is shown that the proposed control approach using both adaptive and conventional TDE parts outperforms a Backstepping-based Integral Sliding Mode Control (BS-ISMC)., Comment: This paper was accepted to 20th International Conference on ADVANCED ROBOTICS 2021

In this paper, we consider the optimal coordination of automated vehicles at intersections under fixed crossing orders. We formulate the problem using direct optimal control and exploit the structure to construct a semi-distributed primal-dual interior-point algorithm to solve it by parallelizing most of the computations. Differently from standard distributed optimization algorithms, where the optimization problem is split, in our approach we split the linear algebra steps, such that the algorithm takes the same steps as a fully centralized one, while still performing computations in a distributed fashion. We analyze the communication requirements of the algorithm, and propose an approximation scheme which can significantly reduce the data exchange. We demonstrate the effectiveness of the algorithm in hard but realistic scenarios, which show that the approximation leads to reductions in communicated data of almost 99\% of the exact formulation, at the expense of less than 1\% suboptimality.

Model predictive control (MPC) is increasingly being considered for control of fast systems and embedded applications. However, the MPC has some significant challenges for such systems. Its high computational complexity results in high power consumption from the control algorithm, which could account for a significant share of the energy resources in battery-powered embedded systems. The MPC parameters must be tuned, which is largely a trial-and-error process that affects the control performance, the robustness and the computational complexity of the controller to a high degree. In this paper, we propose a novel framework in which any parameter of the control algorithm can be jointly tuned using reinforcement learning(RL), with the goal of simultaneously optimizing the control performance and the power usage of the control algorithm. We propose the novel idea of optimizing the meta-parameters of MPCwith RL, i.e. parameters affecting the structure of the MPCproblem as opposed to the solution to a given problem. Our control algorithm is based on an event-triggered MPC where we learn when the MPC should be re-computed, and a dual mode MPC and linear state feedback control law applied in between MPC computations. We formulate a novel mixture-distribution policy and show that with joint optimization we achieve improvements that do not present themselves when optimizing the same parameters in isolation. We demonstrate our framework on the inverted pendulum control task, reducing the total computation time of the control system by 36% while also improving the control performance by 18.4% over the best-performing MPC baseline., Comment: This work has been submitted to the IEEE for possible publication

This paper is concerned with solving chance-constrained finite-horizon optimal control problems, with a particular focus on the recursive feasibility issue of stochastic model predictive control (SMPC) in terms of mission-wide probability of safety (MWPS). MWPS assesses the probability that the entire state trajectory lies within the constraint set, and the objective of the SMPC controller is to ensure that it is no less than a threshold value. This differs from classic SMPC where the probability that the state lies in the constraint set is enforced independently at each time instant. Unlike robust MPC, where strict recursive feasibility is satisfied by assuming that the uncertainty is supported by a compact set, the proposed concept of recursive feasibility for MWPS is based on the notion of remaining MWPSs, which is conserved in the expected value sense. We demonstrate the idea of mission-wide SMPC in the linear SMPC case by deploying a scenario-based algorithm.

Purpose: To evaluate the clinical need for automated decision-support platforms for Adaptive Radiotherapy Therapy (ART) of Head & Neck cancer (HNC) patients. Methods: We tested RTapp (SegAna), a new decision-support software for ART, to investigate 22 HNC patients data retrospectively. For each fraction, RTapp estimated the daily and cumulative doses received by targets and OARs from daily 3D imaging in real-time. RTapp also included a prediction algorithm that analyzed dosimetric parameters (DP) trends against dosimetric endpoints (DE) to trigger adaptation up to 4 fractions ahead. Warning (V95<95%) and adaptation (V95<93%) DEs were set for PTVs. OAR adaptation DEs of +10% (DE10) were set for all Dmax and Dmean DPs. Any DE violation at end of treatment (EOT) triggered a DP trends review to determine the DE-crossing fraction Fx and evaluate the prediction model accuracy (difference between calculated and predicted DP values with 95% confidence intervals). Results: RTapp was able to address the needs of treatment adaptation. 15/22 studies (68%) violated PTV coverage or parotids Dmean at EOT. 9 PTVs had V95<95% (mean coverage decrease of -7.7+/-3.3 %) including 4 flagged for adaptation at median Fx=11.5 (range: 6-18). 15 parotids were flagged for exceeding Dmean constraints with median increase of +3.18 Gy (range: 0.18-6.31 Gy) at EOT, including 8 with DP>DE10. The differences between predicted and calculated PTV V95 and parotids Dmean was up to 7.6% (mean: -2.9+/-4.6 %) and 5 Gy (mean: 0.2+/-1.6 Gy), respectively. The most accurate predictions were obtained closest to Fx. For parotids, Fx ranged between fractions 1 to 23, the lack of specific trend demonstrated the need to verify treatment adaptation for every fraction. Conclusion: Integrated in an ART clinical workflow, RTapp can predict whether specific treatment would require adaptation up to 4 fractions ahead of time., Comment: 20 pages, 7 figures, submitted to Frontiers in Oncology: Radiation Oncology

The cost of the power distribution infrastructures is driven by the peak power encountered in the system. Therefore, the distribution network operators consider billing consumers behind a common transformer in the function of their peak demand and leave it to the consumers to manage their collective costs. This management problem is, however, not trivial. In this paper, we consider a multi-agent residential smart grid system, where each agent has local renewable energy production and energy storage, and all agents are connected to a local transformer. The objective is to develop an optimal policy that minimizes the economic cost consisting of both the spot-market cost for each consumer and their collective peak-power cost. We propose to use a parametric Model Predictive Control (MPC)-scheme to approximate the optimal policy. The optimality of this policy is limited by its finite horizon and inaccurate forecasts of the local power production-consumption. A Deterministic Policy Gradient (DPG) method is deployed to adjust the MPC parameters and improve the policy. Our simulations show that the proposed MPC-based Reinforcement Learning (RL) method can effectively decrease the long-term economic cost for this smart grid problem., Comment: This paper has been accepted to be presented at the 2021 IEEE Conference on Decision and Control (CDC), 6 pages, 6 figures

Economic Model Predictive Control has recently gained popularity due to its ability to directly optimize a given performance criterion, while enforcing constraint satisfaction for nonlinear systems. Recent research has developed both numerical algorithms and stability analysis for the undiscounted case. The introduction of a discount factor in the cost, however, can be desirable in some cases of interest, e.g., economics, stochastically terminating processes, Markov decision processes, etc. Unfortunately, the stability theory in this case is still not fully developed. In this paper we propose a new dissipativity condition to prove asymptotic stability in the infinite horizon case and we connect our results with existing ones in the literature on discounted economic optimal control. Numerical examples are provided to illustrate the theoretical results.

In this work, we propose a Model Predictive Control (MPC)-based Reinforcement Learning (RL) method for Autonomous Surface Vehicles (ASVs). The objective is to find an optimal policy that minimizes the closed-loop performance of a simplified freight mission, including collision-free path following, autonomous docking, and a skillful transition between them. We use a parametrized MPC-scheme to approximate the optimal policy, which considers path-following/docking costs and states (position, velocity)/inputs (thruster force, angle) constraints. The Least Squares Temporal Difference (LSTD)-based Deterministic Policy Gradient (DPG) method is then applied to update the policy parameters. Our simulation results demonstrate that the proposed MPC-LSTD-based DPG method could improve the closed-loop performance during learning for the freight mission problem of ASV., Comment: 6 pages, 7 figures, this paper has been accepted to be presented at 2021 60th IEEE Conference on Decision and Control (CDC)

In the Economic Nonlinear Model Predictive (ENMPC) context, closed-loop stability relates to the existence of a storage function satisfying a dissipation inequality. Finding the storage function is in general -- for nonlinear dynamics and cost -- challenging, and has attracted attentions recently. Q-Learning is a well-known Reinforcement Learning (RL) techniques that attempts to capture action-value functions based on the state-input transitions and stage cost of the system. In this paper, we present the use of the Q-Learning approach to obtain the storage function and verify the dissipativity for discrete-time systems subject to state-input constraints. We show that undiscounted Q-learning is able to capture the storage function for dissipative problems when the parameterization is rich enough. The efficiency of the proposed method will be illustrated in the different case studies., Comment: This paper has been accepted to be presented NMPC2021. 6 pages

Economic Model Predictive Control (MPC) dissipativity theory is central to discussing the stability of policies resulting from minimizing economic stage costs. In its current form, the dissipativity theory for economic MPC applies to problems based on deterministic dynamics or to very specific classes of stochastic problems, and does not readily extend to generic Markov Decision Processes. In this paper, we clarify the core reason for this difficulty, and propose a generalization of the economic MPC dissipativity theory that circumvents it. This generalization focuses on undiscounted infinite-horizon problems and is based on nonlinear stage cost functionals, allowing one to discuss the Lyapunov asymptotic stability of policies for Markov Decision Processes in terms of the probability measures underlying their stochastic dynamics. This theory is illustrated for the stochastic Linear Quadratic Regulator with Gaussian process noise, for which a storage functional can be provided explicitly. For the sake of brevity, we limit our discussion to undiscounted Markov Decision Processes.

We present a Reinforcement Learning-based Robust Nonlinear Model Predictive Control (RL-RNMPC) framework for controlling nonlinear systems in the presence of disturbances and uncertainties. An approximate Robust Nonlinear Model Predictive Control (RNMPC) of low computational complexity is used in which the state trajectory uncertainty is modelled via ellipsoids. Reinforcement Learning is then used in order to handle the ellipsoidal approximation and improve the closed-loop performance of the scheme by adjusting the MPC parameters generating the ellipsoids. The approach is tested on a simulated Wheeled Mobile Robot (WMR) tracking a desired trajectory while avoiding static obstacles., Comment: This paper has been accepted to 2021 European Control Conference (ECC)

In this paper, we are interested in optimal control problems with purely economic costs, which often yield optimal policies having a (nearly) bang-bang structure. We focus on policy approximations based on Model Predictive Control (MPC) and the use of the deterministic policy gradient method to optimize the MPC closed-loop performance in the presence of unmodelled stochasticity or model error. When the policy has a (nearly) bang-bang structure, we observe that the policy gradient method can struggle to produce meaningful steps in the policy parameters. To tackle this issue, we propose a homotopy strategy based on the interior-point method, providing a relaxation of the policy during the learning. We investigate a specific well-known battery storage problem, and show that the proposed method delivers a homogeneous and faster learning than a classical policy gradient approach., Comment: This paper has been accepted to ECC2021. 6 pages

In this paper, we discuss the deterministic policy gradient using the Actor-Critic methods based on the linear compatible advantage function approximator, where the input spaces are continuous. When the policy is restricted by hard constraints, the exploration may not be Centred or Isotropic (non-CI). As a result, the policy gradient estimation can be biased. We focus on constrained policies based on Model Predictive Control (MPC) schemes and to address the bias issue, we propose an approximate Robust MPC approach accounting for the exploration. The RMPC-based policy ensures that a Centered and Isotropic (CI) exploration is approximately feasible. A posterior projection is used to ensure its exact feasibility, we formally prove that this approach does not bias the gradient estimation., Comment: This paper has been accepted to ECC 2021. 6 pages

In this paper, we present the use of Reinforcement Learning (RL) based on Robust Model Predictive Control (RMPC) for the control of an Autonomous Surface Vehicle (ASV). The RL-MPC strategy is utilized for obstacle avoidance and target (set-point) tracking. A scenario-tree robust MPC is used to handle potential failures of the ship thrusters. Besides, the wind and ocean current are considered as unknown stochastic disturbances in the real system, which are handled via constraints tightening. The tightening and other cost parameters are adjusted by RL, using a Q-learning technique. An economic cost is considered, minimizing the time and energy required to achieve the ship missions. The method is illustrated in simulation on a nonlinear 3-DOF model of a scaled version of the Cybership 2., Comment: This paper has been accepted to 2021 American Control Conference (ACC). 6 pages, 9 figures

This paper proposes an observer-based framework for solving Partially Observable Markov Decision Processes (POMDPs) when an accurate model is not available. We first propose to use a Moving Horizon Estimation-Model Predictive Control (MHE-MPC) scheme in order to provide a policy for the POMDP problem, where the full state of the real process is not measured and necessarily known. We propose to parameterize both MPC and MHE formulations, where certain adjustable parameters are regarded for tuning the policy. In this paper, for the sake of tackling the unmodeled and partially observable dynamics, we leverage the Reinforcement Learning (RL) to tune the parameters of MPC and MHE schemes jointly, with the closed-loop performance of the policy as a goal rather than model fitting or the MHE performance. Illustrations show that the proposed approach can effectively increase the performance of close-loop control of systems formulated as POMDPs., Comment: This paper has been accepted to 2021 American Control Conference (ACC) to be held in New Orleans, USA

Model predictive control (MPC) is a powerful trajectory optimization control technique capable of controlling complex nonlinear systems while respecting system constraints and ensuring safe operation. The MPC's capabilities come at the cost of a high online computational complexity, the requirement of an accurate model of the system dynamics, and the necessity of tuning its parameters to the specific control application. The main tunable parameter affecting the computational complexity is the prediction horizon length, controlling how far into the future the MPC predicts the system response and thus evaluates the optimality of its computed trajectory. A longer horizon generally increases the control performance, but requires an increasingly powerful computing platform, excluding certain control applications.The performance sensitivity to the prediction horizon length varies over the state space, and this motivated the adaptive horizon model predictive control (AHMPC), which adapts the prediction horizon according to some criteria. In this paper we propose to learn the optimal prediction horizon as a function of the state using reinforcement learning (RL). We show how the RL learning problem can be formulated and test our method on two control tasks, showing clear improvements over the fixed horizon MPC scheme, while requiring only minutes of learning., Comment: This work has been submitted to IFAC NMPC 2021 for possible publication

In this paper, we consider solving discounted Markov Decision Processes (MDPs) under the constraint that the resulting policy is stabilizing. In practice MDPs are solved based on some form of policy approximation. We will leverage recent results proposing to use Model Predictive Control (MPC) as a structured policy in the context of Reinforcement Learning to make it possible to introduce stability requirements directly inside the MPC-based policy. This will restrict the solution of the MDP to stabilizing policies by construction. The stability theory for MPC is most mature for the undiscounted MPC case. Hence, we will first show in this paper that stable discounted MDPs can be reformulated as undiscounted ones. This observation will entail that the MPC-based policy with stability requirements will produce the optimal policy for the discounted MDP if it is stable, and the best stabilizing policy otherwise.

The combination of learning methods with Model Predictive Control (MPC) has attracted a significant amount of attention in the recent literature. The hope of this combination is to reduce the reliance of MPC schemes on accurate models, and to tap into the fast developing machine learning and reinforcement learning tools to exploit the growing amount of data available for many systems. In particular, the combination of reinforcement learning and MPC has been proposed as a viable and theoretically justified approach to introduce explainable, safe and stable policies in reinforcement learning. However, a formal theory detailing how the safety and stability of an MPC-based policy can be maintained through the parameter updates delivered by the learning tools is still lacking. This paper addresses this gap. The theory is developed for the generic Robust MPC case, and applied in simulation in the robust tube-based linear MPC case, where the theory is fairly easy to deploy in practice. The paper focuses on Reinforcement Learning as a learning tool, but it applies to any learning method that updates the MPC parameters online.

In control applications there is often a compromise that needs to be made with regards to the complexity and performance of the controller and the computational resources that are available. For instance, the typical hardware platform in embedded control applications is a microcontroller with limited memory and processing power, and for battery powered applications the control system can account for a significant portion of the energy consumption. We propose a controller architecture in which the computational cost is explicitly optimized along with the control objective. This is achieved by a three-part architecture where a high-level, computationally expensive controller generates plans, which a computationally simpler controller executes by compensating for prediction errors, while a recomputation policy decides when the plan should be recomputed. In this paper, we employ model predictive control (MPC) as the high-level plan-generating controller, a linear state feedback controller as the simpler compensating controller, and reinforcement learning (RL) to learn the recomputation policy. Simulation results for two examples showcase the architecture's ability to improve upon the MPC approach and find reasonable compromises weighing the performance on the control objective and the computational resources expended., Comment: Submitted to 3rd Annual Learning for Dynamics and Control Conference (L4DC 2021)

The main contribution of this paper is a novel method for planning globally optimal trajectories for dynamical systems subject to polygonal constraints. The proposed method is a hybrid trajectory planning approach, which combines graph search, i.e. a discrete roadmap method, with convex optimization, i.e. a complete path method. Contrary to past approaches, which have focused on using simple obstacle approximations, or sub-optimal spatial discretizations, our approach is able to use the exact geometry of polygonal constraints in order to plan optimal trajectories. The performance and flexibility of the proposed method is evaluated via simulations by planning distance-optimal trajectories for a Dubins car model, as well as time-, distance- and energy-optimal trajectories for a marine vehicle.

Reinforcement Learning (RL) has proven a stunning ability to learn optimal policies from data without any prior knowledge on the process. The main drawback of RL is that it is typically very difficult to guarantee stability and safety. On the other hand, Nonlinear Model Predictive Control (NMPC) is an advanced model-based control technique which does guarantee safety and stability, but only yields optimality for the nominal model. Therefore, it has been recently proposed to use NMPC as a function approximator within RL. While the ability of this approach to yield good performance has been demonstrated, the main drawback hindering its applicability is related to the computational burden of NMPC, which has to be solved to full convergence. In practice, however, computationally efficient algorithms such as the Real-Time Iteration (RTI) scheme are deployed in order to return an approximate NMPC solution in very short time. In this paper we bridge this gap by extending the existing theoretical framework to also cover RL based on RTI NMPC. We demonstrate the effectiveness of this new RL approach with a nontrivial example modeling a challenging nonlinear system subject to stochastic perturbations with the objective of optimizing an economic cost., Comment: accepted for the IFAC World Congress 2020

In this paper we propose and compare methods for combining system identification (SYSID) and reinforcement learning (RL) in the context of data-driven model predictive control (MPC). Assuming a known model structure of the controlled system, and considering a parametric MPC, the proposed approach simultaneously: a) Learns the parameters of the MPC using RL in order to optimize performance, and b) fits the observed model behaviour using SYSID. Six methods that avoid conflicts between the two optimization objectives are proposed and evaluated using a simple linear system. Based on the simulation results, hierarchical, parallel projection, nullspace projection, and singular value projection achieved the best performance., Comment: Accepted to the IFAC2020