Sébastien Gros

Wenqi Cai, Kyriakos G. Vamvoudakis, Sébastien Gros et al.

In this paper, we propose a cost-matching approach for optimal humanoid locomotion within a Model Predictive Control (MPC)-based Reinforcement Learning (RL) framework. A parameterized MPC formulation with centroidal dynamics is trained to approximate the action-value function obtained from high-fidelity closed-loop data. Specifically, the MPC cost-to-go is evaluated along recorded state-action trajectories, and the parameters are updated to minimize the discrepancy between MPC-predicted values and measured returns. This formulation enables efficient gradient-based learning while avoiding the computational burden of repeatedly solving the MPC problem during training. The proposed method is validated in simulation using a commercial humanoid platform. Results demonstrate improved locomotion performance and robustness to model mismatch and external disturbances compared with manually tuned baselines.

Huang Zhang, Yang Su, Faisal Altaf et al.

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.

Erlend Torje Berg Lundby, Adil Rasheed, Ivar Johan Halvorsen et al.

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.

Shambhuraj Sawant, Akhil S Anand, Dirk Reinhardt et al.

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.

Arash Bahari Kordabad, Hossein Nejatbakhsh Esfahani, Wenqi Cai et al.

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.

Shambhuraj Sawant, Sebastien Gros

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.

Bowen Song, Sebastien Gros, Andrea Iannelli

Policy gradient algorithms are widely used in reinforcement learning and belong to the class of approximate dynamic programming methods. This paper studies two key policy gradient algorithms, the Natural Policy Gradient and the Gauss-Newton Method, for solving the Linear Quadratic Regulator (LQR) problem in unknown stochastic linear systems. The main challenge lies in obtaining an unbiased gradient estimate from noisy data due to errors-in-variables in linear regression. This issue is addressed by employing a primal-dual estimation procedure. Using this novel gradient estimation scheme, the paper establishes convergence guarantees with a sample complexity of order O(1/epsilon). Theoretical results are further supported by numerical experiments, which demonstrate the effectiveness of the proposed algorithms.

Jonathan Frey, Katrin Baumgärtner, Gianluca Frison et al.

The efficient computation of parametric solution sensitivities is a key challenge in the integration of learning-enhanced methods with nonlinear model predictive control (MPC), as their availability is crucial for many learning algorithms. While approaches presented in the machine learning community are limited to convex or unconstrained formulations, this paper discusses the computation of solution sensitivities of general nonlinear programs (NLPs) using the implicit function theorem (IFT) and smoothed optimality conditions treated in interior-point methods (IPM). We detail sensitivity computation within a sequential quadratic programming (SQP) method which employs an IPM for the quadratic subproblems. The publication is accompanied by an efficient open-source implementation within the framework, providing both forward and adjoint sensitivities for general optimal control problems, achieving speedups exceeding 3x over the state-of-the-art solver mpc.pytorch.

Arash Bahari Kordabad, Dean Brandner, Sebastien Gros et al.

In this paper, we propose a second-order deterministic actor-critic framework in reinforcement learning that extends the classical deterministic policy gradient method to exploit curvature information of the performance function. Building on the concept of compatible function approximation for the critic, we introduce a quadratic critic that simultaneously preserves the true policy gradient and an approximation of the performance Hessian. A least-squares temporal difference learning scheme is then developed to estimate the quadratic critic parameters efficiently. This construction enables a quasi-Newton actor update using information learned by the critic, yielding faster convergence compared to first-order methods. The proposed approach is general and applicable to any differentiable policy class. Numerical examples demonstrate that the method achieves improved convergence and performance over standard deterministic actor-critic baselines.

Rudolf Reiter, Jasper Hoffmann, Dirk Reinhardt et al.

The fields of MPC and RL consider two successful control techniques for Markov decision processes. Both approaches are derived from similar fundamental principles, and both are widely used in practical applications, including robotics, process control, energy systems, and autonomous driving. Despite their similarities, MPC and RL follow distinct paradigms that emerged from diverse communities and different requirements. Various technical discrepancies, particularly the role of an environment model as part of the algorithm, lead to methodologies with nearly complementary advantages. Due to their orthogonal benefits, research interest in combination methods has recently increased significantly, leading to a large and growing set of complex ideas leveraging MPC and RL. This work illuminates the differences, similarities, and fundamentals that allow for different combination algorithms and categorizes existing work accordingly. Particularly, we focus on the versatile actor-critic RL approach as a basis for our categorization and examine how the online optimization approach of MPC can be used to improve the overall closed-loop performance of a policy.

Esmaeel Mohammadi, Daniel Ortiz-Arroyo, Aviaja Anna Hansen et al.

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.

Akhil S Anand, Shambhuraj Sawant, Dirk Reinhardt et al.

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.

Akhil S Anand, Shambhuraj Sawant, Jasper Hoffmann et al.

Sim2Real aims at training policies in high-fidelity simulation environments and effectively transferring them to the real world. Despite the developments of accurate simulators and Sim2Real RL approaches, the policies trained purely in simulation often suffer significant performance drops when deployed in real environments. This drop is referred to as the Sim2Real performance gap. Current Sim2Real RL methods optimize the simulator accuracy and variability as proxies for real-world performance. However, these metrics do not necessarily correlate with the real-world performance of the policy as established theoretically and empirically in the literature. We propose a novel framework to address this issue by directly adapting the simulator parameters based on real-world performance. We frame this problem as a bi-level RL framework: the inner-level RL trains a policy purely in simulation, and the outer-level RL adapts the simulation model and in-sim reward parameters to maximize real-world performance of the in-sim policy. We derive and validate in simple examples the mathematical tools needed to develop bi-level RL algorithms that close the Sim2Real performance gap.

Hannah Markgraf, Shamburaj Sawant, Hanna Krasowski et al.

Projection-based safety filters, which modify unsafe actions by mapping them to the closest safe alternative, are widely used to enforce safety constraints in reinforcement learning (RL). Two integration strategies are commonly considered: Safe environment RL (SE-RL), where the safeguard is treated as part of the environment, and safe policy RL (SP-RL), where it is embedded within the policy through differentiable optimization layers. Despite their practical relevance in safety-critical settings, a formal understanding of their differences is lacking. In this work, we present a theoretical comparison of SE-RL and SP-RL. We identify a key distinction in how each approach is affected by action aliasing, a phenomenon in which multiple unsafe actions are projected to the same safe action, causing information loss in the policy gradients. In SE-RL, this effect is implicitly approximated by the critic, while in SP-RL, it manifests directly as rank-deficient Jacobians during backpropagation through the safeguard. Our contributions are threefold: (i) a unified formalization of SE-RL and SP-RL in the context of actor-critic algorithms, (ii) a theoretical analysis of their respective policy gradient estimates, highlighting the role of action aliasing, and (iii) a comparative study of mitigation strategies, including a novel penalty-based improvement for SP-RL that aligns with established SE-RL practices. Empirical results support our theoretical predictions, showing that action aliasing is more detrimental for SP-RL than for SE-RL. However, with appropriate improvement strategies, SP-RL can match or outperform improved SE-RL across a range of environments. These findings provide actionable insights for choosing and refining projection-based safe RL methods based on task characteristics.

Dean Brandner, Sebastien Gros, Sergio Lucia

Model predictive control (MPC) is widely used in process control due to its interpretability and ability to handle constraints. As a parametric policy in reinforcement learning (RL), MPC offers strong initial performance and low data requirements compared to black-box policies like neural networks. However, most RL methods rely on first-order updates, which scale well to large parameter spaces but converge at most linearly, making them inefficient when each policy update requires solving an optimal control problem, as is the case with MPC. While MPC policies are typically sparsely parameterized and thus amenable to second-order approaches, existing second-order methods demand second-order policy derivatives, which can be computationally and memory-wise intractable. This work introduces a Gauss-Newton approximation of the deterministic policy Hessian that eliminates the need for second-order policy derivatives, enabling superlinear convergence with minimal computational overhead. To further improve robustness, we propose a momentum-based Hessian averaging scheme for stable training under noisy estimates. We demonstrate the effectiveness of the approach on a nonlinear continuously stirred tank reactor (CSTR), showing faster convergence and improved data efficiency over state-of-the-art first-order methods.

Eivind Bøhn, Sebastien Gros, Signe Moe et al.

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.

Hossein Nejatbakhsh Esfahani, Arash Bahari Kordabad, Sebastien Gros

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.

Arash Bahari Kordabad, Wenqi Cai, Sebastien Gros

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.

Eivind Bøhn, Sebastien Gros, Signe Moe et al.

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.

Mario Zanon, Sébastien Gros, Michele Palladino

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.

Sébastien Gros, Mario Zanon

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.

Eivind Bøhn, Sebastien Gros, Signe Moe et al.

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.

Sebastien Gros, Mario Zanon

Model Predictive Control has been recently proposed as policy approximation for Reinforcement Learning, offering a path towards safe and explainable Reinforcement Learning. This approach has been investigated for Q-learning and actor-critic methods, both in the context of nominal Economic MPC and Robust (N)MPC, showing very promising results. In that context, actor-critic methods seem to be the most reliable approach. Many applications include a mixture of continuous and integer inputs, for which the classical actor-critic methods need to be adapted. In this paper, we present a policy approximation based on mixed-integer MPC schemes, and propose a computationally inexpensive technique to generate exploration in the mixed-integer input space that ensures a satisfaction of the constraints. We then propose a simple compatible advantage function approximation for the proposed policy, that allows one to build the gradient of the mixed-integer MPC-based policy.

Sebastien Gros, Mario Zanon, Alberto Bemporad

For all its successes, Reinforcement Learning (RL) still struggles to deliver formal guarantees on the closed-loop behavior of the learned policy. Among other things, guaranteeing the safety of RL with respect to safety-critical systems is a very active research topic. Some recent contributions propose to rely on projections of the inputs delivered by the learned policy into a safe set, ensuring that the system safety is never jeopardized. Unfortunately, it is unclear whether this operation can be performed without disrupting the learning process. This paper addresses this issue. The problem is analysed in the context of $Q$-learning and policy gradient techniques. We show that the projection approach is generally disruptive in the context of $Q$-learning though a simple alternative solves the issue, while simple corrections can be used in the context of policy gradient methods in order to ensure that the policy gradients are unbiased. The proposed results extend to safe projections based on robust MPC techniques.

Mario Zanon, Sébastien Gros, Alberto Bemporad

Reinforcement Learning (RL) has demonstrated a huge potential in learning optimal policies without any prior knowledge of the process to be controlled. Model Predictive Control (MPC) is a popular control technique which is able to deal with nonlinear dynamics and state and input constraints. The main drawback of MPC is the need of identifying an accurate model, which in many cases cannot be easily obtained. Because of model inaccuracy, MPC can fail at delivering satisfactory closed-loop performance. Using RL to tune the MPC formulation or, conversely, using MPC as a function approximator in RL allows one to combine the advantages of the two techniques. This approach has important advantages, but it requires an adaptation of the existing algorithms. We therefore propose an improved RL algorithm for MPC and test it in simulations on a rather challenging example.