While Model Predictive Control (MPC) is a widely used method for controlling constrained nonlinear systems, data-driven MPC has recently emerged as a viable alternative to explicit MPC when sufficient data is available. This work investigates two neural network-based strategies for learning control policies in nonlinear systems: supervised and unsupervised learning. The supervised strategy approximates the MPC feedback law using a neural network trained on demonstrations generated from offline MPC simulations. In contrast, the unsupervised approach directly minimizes an MPC-inspired cost function using a learned dynamical model, avoiding the need for expert-generated trajectories. Beyond empirical evaluation, the supervised approach is complemented by a generalization analysis that yields generalization guarantees for a simplified neural architecture and motivates an early-stopping strategy. To retain constraint handling when replacing the online MPC optimization with a compact neural policy, a one-step feasibility filter is introduced that projects the network output onto the admissible set through a small nonlinear program. The proposed framework is evaluated numerically on a realistic simulation of an open-die forging process with process and measurement noise, where deformation speed must be finely regulated to ensure product homogeneity. Simulation results show that both learning-based controllers achieve MPC-level tracking performance with compact networks, remain robust under disturbances, and reduce online computation time by more than an order of magnitude. (c) 2026 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

The early stopping strategy consists in stopping the training process of a neural network (NN) on a set S of input data before training error is minimal. The advantage is that the NN then retains good generalization properties, i.e. it gives good predictions on data outside S, and a good estimate of the statistical error ("population loss") is obtained. Using the theories of Rademacher complexity and neural tangent kernel, we give here two stopping strategies that minimize upper bounds on the population loss. These methods are wellsuited to the underparameterized context (where the number of parameters is moderate compared with the number of data). They are illustrated on the example of an NN simulating the model predictive control of a Van der Pol oscillator.

Model Predictive Control (MPC) is a traditional technique widely employed to control constrained nonlinear systems. Recently, data-driven MPC has emerged as an alternative to explicit MPC strategies when sufficient data are available. However, there has been limited progress in approximating nonlinear MPC to alleviate the computational burden for realtime applications while ensuring constraint satisfaction. In this paper, we use a feed-forward neural network to approximate a classical MPC controller, thereby reducing computational complexity. To guarantee constraint satisfaction, we project the network’s prediction onto a control invariant set. We apply the proposed strategy in a simulation of an open-die forging process, which is highly nonlinear and prone to delays.

In this paper, we consider the problem of computing from data guaranteed set-valued over-approximations of unknown monotone functions with additive disturbances. We provide a characterization of a simulating map that provably contains all monotone functions that are consistent with the data. This map is also minimal in the sense that any set-valued map containing all consistent monotone functions would also include the map we are proposing. We show that this minimal simulating map is interval-valued and admits a simple construction on a finite partition induced by the data. As the complexity of the partition increases with the amount of data, we also consider the problem of computing minimal interval-valued simulating maps defined on partitions that are fixed a priori. We present an efficient algorithm for their computation. We then use those data-driven over-approximations to build models for partially unknown systems where the unknown part is monotone. The resulting models are used to construct finite-state symbolic abstractions, paving the way for discrete controller synthesis methods to be applied. We extend our approach to handle systems with bounded derivatives and introduce an algorithm to calculate the bounds on those derivatives and on the disturbances from the data. We present several numerical experiments to test the performance of the introduced method and show that the data-driven abstractions are suitable for controller synthesis purposes.

In this paper, we introduce an online safe learning -based Model Predictive Control (MPC) approach. This approach, which we call the "compatible model approach", consists of building two models of the system. The first is a piecewise interval-valued over-approximation of the system, and the second is a single-valued piecewise multi-affine estimation of the system's dynamics The first model is used to find the set of safe actions at each state, whereas the second is used to choose -out of those safe actions- the input that minimizes a given cost function. For the first model, we use the -assumed known- bounds on the derivative of the dynamics to update the model. The second model should be contained in the first to ensure the feasibility of the MPC scheme everywhere (Hence, the name compatible). Both models can be updated online. We are able to do that because each new transition updates the models locally. We present a test case where we train a mobile robot at low speeds, then navigate it in an environment while avoiding obstacles and collecting new data to learn its dynamics at high speeds. Copyright (c) 2024 The Authors.

In this paper, we introduce a novel approach to safe learning-based Model Predictive Control (MPC) for nonlinear systems. This approach, which we call the “compatible model approach”, relies on computing two models of the given unknown system using data generated from the system. The first model is a set-valued over-approximation guaranteed to contain the system’s dynamics. This model is used to find a set of provably safe controller actions at every state. The second model is a single-valued estimation of the system’s dynamics used to find a controller that minimises a cost function. If the two models are compatible, in the sense that the estimation is included in the over-approximation, we show that we can use the set of safe controller actions to constrain the minimisation problem and guarantee the feasibility and safety of the learning-based MPC controller at all times. We present a method to build an over-approximation for nonlinear systems with bounded derivatives on a partition of the states and inputs spaces. Then, we use piecewise multi-affine functions (defined on the same partition) to calculate a system’s dynamics estimation that is compatible with the previous over-approximation. Finally, we show the effectiveness of the approach by considering a path-planning problem with obstacle avoidance.

It has been theoretically explained, through the notion of Neural Tangent Kernel, why the training error of overparameterized networks converges linearly to 0. In this letter, we focus on the case of small (or underparameterized) networks. An advantage of small networks is that they are faster to train while retaining sufficient precision to perform useful tasks in many applications. Our main theoretical contribution is to prove that the training error of small networks converges linearly to a (non-null) constant, of which we give a precise estimate. We verify this result on a neural network of 10 neurons simulating a Model Predictive Controller. We also observe that an upper bound of the generalization error follows a double-peak curve as the number of training data increases.

Methods based on "(Jacobian) matrix measure" to show the convergence of a dynamical system to a limit cycle (LC), generally assume that the measure is negative everywhere on the LC. We relax this assumption by assuming that the matrix measure is negative "on average" over one period of LC. Using an approximate Euler trajectory, we thus present a method that guarantees the LC existence, and allows us to construct a basin of attraction. This is illustrated on the example of the Van der Pol system.

It was shown in the literature that, for a fully connected neural network (NN), the gradient descent algorithm converges to zero. Motivated by that work, we provide here general conditions under which we can derive the convergence of the gradient descent algorithm from the convergence of the gradient flow, in the case of NNs, in a systematic way. Our approach is based on an analysis of the error in Euler's method in the case of NNs, and relies on the concept of local strong convexity. Unlike existing approaches in the literature, our approach allows to provide convergence guarantees without making any assumptions on the number of hidden nodes of the NN or the number of training data points. A numerical example is proposed, showing the merits of our approach.

In the gradient descent method, one often focus on the convergence of the sequence generated by the algorithm, but less often on the deviation of these points from the solutions of the original continuous -time differential equation (gradient flow). This also happens when discretizing other ordinary differential equations. In the case of a discretization by explicit Euler's method with a constant step h , we provide here sufficient conditions, in terms of strong monotonicity and co-coercivity, for the deviation between discrete and continuous solutions to tend asymptotically towards zero. This analysis could shed new light on some applications of the gradient descent algorithm. (c) 2022 European Control Association. Published by Elsevier Ltd. All rights reserved.

This article presents a symbolic control approach to the design of distributed safety controllers for a class of continuous-time nonlinear systems. More precisely, we consider systems made of components where each component is equipped with a sampled-data controller with its own sampling period, resulting globally in a distributed multiperiodic sampled-data system. Moreover, controllers receive partial information on the state of the other components. We propose a component-based approach to controller synthesis, which relies on the use of abstractions and continuous-time assume-guarantee contracts. The abstractions describe the dynamics of the system from the point of view of each component based on the information structure, whereas assume-guarantee contracts specify guarantees that a component must satisfy if assumptions on the other components are met. We show that our approach makes it possible to decompose a global safety control problem into local ones that can be solved independently. We then show how symbolic control techniques can be used to synthesize controllers that enforce the local control objectives. Illustrative applications in building automation and vehicle platooning are shown.

Given a dynamical system Σp with a parameter p taking its values in a fixed interval Q, we present a simple criterion of set inclusion which guarantees that the Euler approximate solutions of Σpo for some value po ∈ Q converge to a limit cycle E. Moreover, we characterize a compact set I containing ε which is invariant for the exact solutions of Σp whatever the value of p ∈ Q. We illustrate the application of our method on the example of a parametric Van der Pol system driven by a periodic input.

In this paper, we present a novel approach for the abstraction of monotone systems with bounded disturbances. The approach is data-driven and uses a given set of samples of the (unknown) dynamics of the system to compute an abstraction defined on partitions of the state and input spaces. The proposed method is efficient as its computational complexity is linear in the number of samples and in the size of the partitions. Moreover, the abstraction is shown to be minimally conservative in the absence of disturbances. We show that the resulting symbolic model is itself a monotone transition system and is related to the original system by an alternating simulation relation. We present some numerical experiments to show the effectiveness of the approach and to show how the choice of the partitions or the number of samples affects the quality of the approximation.

We show here how, using Euler's integration method and an associated function bounding the error in function of time, one can generate structures closely surrounding the invariant tori of dynamical systems. Such structures are constructed from a finite number of balls of $\mathbb{R}^n$ and encompass the deformations of the tori when small perturbations of the flow of the system occur.

This paper focuses on the guaranteed identification of viscous friction parameters for a nonlinear inverted pendulum. The method is based on the interval analysis (IA) and set-inversion tools to determine the set of all the feasible friction parameters from a prior domain of interest, i.e. initial interval vector or box, that are consistent with all the experimental and theoretical datasets including their uncertainties. The capabilities of our proposed guaranteed identification are compared with the more commonly used approach based on the least square method identification (LSMI), which is used especially to adjust the inertial and geometric parameters of our experimental plant. Both of them have been investigated through several experiments on a real inverted pendulum and simulations with uncertain ODEs via the DynIbex library.

In this paper, we introduce an approach for the data-driven abstraction of monotone dynamical systems. First, we introduce a set-valued simulating map, which over-approximates the dynamics of an unknown monotone system, using only a set of transitions generated by it. We establish the minimality of the introduced simulating map. Then, we show that the system, with this map as its transition relation, is equivalent (in the sense of alternating bisimulation) to a finite-state system. This equivalence enables the use of well-established symbolic control techniques to synthesize controllers. We show the effectiveness of the approach on a safety controller synthesis problem.

The next generation of space systems will have to achieve more and more complex missions. In order to master the development cost and duration of such systems, an alternative to a manual design is to automatically synthesize the main parameters of the system. In this paper, we present an approach on the specific case of the scheduling of the flight control of a space launcher. The approach requires two successive steps: (1) the formalization of the problem to be solved in a parametric formal model and (2) the synthesis of the model parameters with a tool. We first describe the problematic of the scheduling of a launcher flight control, then we show how this problematic can be formalized with parametric stopwatch automata; we then present the results computed by IMITATOR. We compare the results to the ones obtained by other tools classically used in scheduling.

We aim at the synthesis of an approximate minimax control for a system dynamic given in the form x(t) = f(x(t), u (t), d(t)) where x represents the state, u the control (or input) and d a disturbance (or perturbation). The disturbance d(t) is prescribed to a compact domain D on which it can take any value (bounded uncertainty). Our method makes use of symbolic computation to enclose the solutions corresponding to all possible disturbance d (-) ϵ D, together with a simple algorithm of random sampling to select the minimum control u⁎(•). We illustrate the interest of this approximate minimax control on an example of a biochemical reactor.