Constrained optimization is popularly seen in reinforcement learning for addressing complex control tasks. From the perspective of dynamic system, iteratively solving a constrained optimization problem can be framed as the temporal evolution of a feedback control system. Classical constrained optimization methods, such as penalty and Lagrangian approaches, inherently use proportional and integral feedback controllers. In this paper, we propose a more generic equivalence framework to build the connection between constrained optimization and feedback control system, for the purpose of developing more effective constrained RL algorithms. Firstly, we define that each step of the system evolution determines the Lagrange multiplier by solving a multiplier feedback optimal control problem (MFOCP). In this problem, the control input is multiplier, the state is policy parameters, the dynamics is described by policy gradient descent, and the objective is to minimize constraint violations. Then, we introduce a multiplier guided policy learning (MGPL) module to perform policy parameters updating. And we prove that the resulting optimal policy, achieved through alternating MFOCP and MGPL, aligns with the solution of the primal constrained RL problem, thereby establishing our equivalence framework. Furthermore, we point out that the existing PID Lagrangian is merely one special case within our framework that utilizes a PID controller. We also accommodate the integration of other various feedback controllers, thereby facilitating the development of new algorithms. As a representative, we employ model predictive control (MPC) as the feedback controller and consequently propose a new algorithm called predictive Lagrangian optimization (PLO). Numerical experiments demonstrate its superiority over the PID Lagrangian method, achieving a larger feasible region up to 7.2% and a comparable average reward.
Inspired greatly by Mills et al. (2009) and the solution within, this paper aims to more clearly explain the mathematics and implementation details of such a powerful control algorithm. While the aforementioned paper is well written and of sound mathematics, it is extreamly dense and requires some time and patience to decipher, especially as it draws on many other sources to complete the algorithm. This dense property is a clear result of the paper being restricted to the brief form and important details being ommited as a result. We provide the much needed elaboration here for the benifit of the reader.
This work presents a robust Model Predictive Controller (MPC) to solve the problem of spacecraft rendezvous in the context of the restricted three-body problem (R3BP) as will be required to dock with space stations in cislunar space. The employed methodology is both valid for chemical and electric thrusters. By exploiting the state transition matrix and using a chance-constrained approach, the robust MPC assures constraints satisfaction under the presence of disturbances in a probabilistic sense. The perturbations parameters are computed on-line using a disturbance estimator. The robust controller is tested for a rendezvous scenario with a target placed in an Earth-Moon Near-Rectilinear Halo Orbit. Numerical results are shown and discussed.
This paper presents a time-constrained model predictive control strategy for the six degree-of-freedom autonomous rendezvous, proximity, operations and docking problem between a controllable "deputy" satellite and an uncontrolled "chief" satellite. The objective is to achieve a docking configuration defined by both the translational and attitudinal states of the deputy relative to the chief, whose dynamics are respectively governed by both the Clohessy-Wiltshire equations and Euler's second law of motion. The proposed control strategy explicitly addresses computational time constraints that are common to state-of-the-art space vehicles. Thus, a time-constrained model predictive control strategy is implemented on a space-grade processor. Although suboptimal with regards to energy consumption when compared to conventional optimal RPO trajectories, it is empirically demonstrated via numerical simulations that the deputy spacecraft still achieves a successful docking configuration while subject to computational time constraints.
The microgrid concept offers high flexibility and resilience due to the possibility of switching between grid-connected and stand-alone operation. This renders microgrids an auspicious solution for rural areas and critical infrastructure. In standalone or islanded mode, the main objective is cost minimization while ensuring a safe and reliable operation. Optimal operation schemes for microgrids usually assume fixed power limits for energy storage units. This, however, is not sufficient for lithiumion energy storage systems, which often come with dynamic power limits that depend on the state of charge. These limits are especially prominent when the state of charge is close to its boundaries. In this paper, dynamic constraints for energy storages are modelled using convex polytopes and fitted to experimental data acquired from an 11.6 kWh lithium-ion energy storage system. The polytopic constraints are integrated in a model predictive control scheme that was designed for a standalone microgrid composed of a fuel cell, a photovoltaic generator and a lithium-ion energy storage system. To evaluate the advantages, a case study with two configurations is performed. The model predictive controller without polytopic constraints led to constraint violations in 11.77 % of the simulation time steps with a maximum deviation of 118 % above the power limits. The configuration with polytopic constraints in contrary led to no violations over the entire simulation horizon.
This paper introduces a new approach that leverages Multi-agent Bayesian Optimization (MABO) to design Distributed Model Predictive Control (DMPC) schemes for multi-agent systems. The primary objective is to learn optimal DMPC schemes even when local model predictive controllers rely on imperfect local models. The proposed method invokes a dual decomposition-based distributed optimization framework, incorporating an Alternating Direction Method of Multipliers (ADMM)-based MABO algorithm to enable coordinated learning of parameterized DMPC schemes. This enhances the closed-loop performance of local controllers, despite discrepancies between their models and the actual multi-agent system dynamics. In addition to the newly proposed algorithms, this work also provides rigorous proofs establishing the optimality and convergence of the underlying learning method. Finally, numerical examples are given to demonstrate the efficacy of the proposed MABO-based learning approach.
Large clinical evidence acknowledges the crucial role played by physical activity in delaying the progression of type-2 diabetes. However, the literature lacks control approaches that leverage exercise for type-2 diabetes control and more in general lacks a quantitative assessment of medical guidelines on the recommended amount of physical activity to be performed, mainly due to the absence of mathematical models that suitably estimate its benefits on diabetes progression. In this work, in order to provide a control-theoretical formulation of the exercise, we design a feedback law in terms of recommended physical activity, following a model predictive control approach, based on a widespread compact diabetes progression model, suitably modified to properly account for the long-term effect of the exercise. Moreover we illustrate how the proposed approach proves to show reliable robustness properties with respect to initial conditions and parameter perturbations, which may be used to reflect inter-patient variability. Results are encouraging in view of the validation of the control law on comprehensive high-dimensional models of diabetes progression, with the aim of translating the prediction of the controller into reasonable recommendations and to quantitatively support medical decision-making.
Model predictive control (MPC) controller is considered for temperature management in buildings but its performance heavily depends on hyperparameters. Consequently, MPC necessitates meticulous hyperparameter tuning to attain optimal performance under diverse contracts. However, conventional building controller design is an open-loop process without critical hyperparameter optimization, often leading to suboptimal performance due to unexpected environmental disturbances and modeling errors. Furthermore, these hyperparameters are not adapted to different pricing schemes and may lead to non-economic operations. To address these issues, we propose an efficient performance-oriented building MPC controller tuning method based on a cutting-edge efficient constrained Bayesian optimization algorithm, CONFIG, with global optimality guarantees. We demonstrate that this technique can be applied to efficiently deal with real-world DSM program selection problems under customized black-box constraints and objectives. In this study, a simple MPC controller, which offers the advantages of reduced commissioning costs, enhanced computational efficiency, was optimized to perform on a comparable level to a delicately designed and computationally expensive MPC controller. The results also indicate that with an optimized simple MPC, the monthly electricity cost of a household can be reduced by up to 26.90% compared with the cost when controlled by a basic rule-based controller under the same constraints. Then we compared 12 real electricity contracts in Belgium for a household family with customized black-box occupant comfort constraints. The results indicate a monthly electricity bill saving up to 20.18% when the most economic contract is compared with the worst one, which again illustrates the significance of choosing a proper electricity contract.
Digital Twins, virtual replicas of physical systems that enable real-time monitoring, model updates, predictions, and decision-making, present novel avenues for proactive control strategies for autonomous systems. However, achieving real-time decision-making in Digital Twins considering uncertainty necessitates an efficient uncertainty quantification (UQ) approach and optimization driven by accurate predictions of system behaviors, which remains a challenge for learning-based methods. This paper presents a simultaneous multi-step robust model predictive control (MPC) framework that incorporates real-time decision-making with uncertainty awareness for Digital Twin systems. Leveraging a multistep ahead predictor named Time-Series Dense Encoder (TiDE) as the surrogate model, this framework differs from conventional MPC models that provide only one-step ahead predictions. In contrast, TiDE can predict future states within the prediction horizon in a one-shot, significantly accelerating MPC. Furthermore, quantile regression is employed with the training of TiDE to perform flexible while computationally efficient UQ on data uncertainty. Consequently, with the deep learning quantiles, the robust MPC problem is formulated into a deterministic optimization problem and provides a safety buffer that accommodates disturbances to enhance constraint satisfaction rate. As a result, the proposed method outperforms existing robust MPC methods by providing less-conservative UQ and has demonstrated efficacy in an engineering case study involving Directed Energy Deposition (DED) additive manufacturing. This proactive while uncertainty-aware control capability positions the proposed method as a potent tool for future Digital Twin applications and real-time process control in engineering systems.
Periodic dynamical systems, distinguished by their repetitive behavior over time, are prevalent across various engineering disciplines. In numerous applications, particularly within industrial contexts, the implementation of model predictive control (MPC) schemes tailored to optimize specific economic criteria was shown to offer substantial advantages. However, the real-time implementation of these schemes is often infeasible due to limited computational resources. To tackle this problem, we propose a resource-efficient economic model predictive control scheme for periodic systems, leveraging existing single-layer MPC techniques. Our method relies on a single quadratic optimization problem, which ensures high computational efficiency for real-time control in dynamic settings. We prove feasibility, stability and convergence to optimum of the proposed approach, and validate the effectiveness through numerical experiments.
Koopman-based lifted linear identification have been widely used for data-driven prediction and model predictive control (MPC) of nonlinear systems. It has found applications in flow-control, soft robotics, and unmanned aerial vehicles (UAV). For autonomous systems, this system identification method works by embedding the nonlinear system in a higher-dimensional linear space and computing a finite-dimensional approximation of the corresponding Koopman operator with the Extended Dynamic Mode Decomposition (EDMD) algorithm. EDMD is a data-driven algorithm that estimates an approximate linear system by lifting the state data-snapshots via nonlinear dictionary functions. For control systems, EDMD is further modified to utilize both state and control data-snapshots to estimate a lifted linear predictor with control input. This article investigates how the estimation process is affected when the data is quantized. Specifically, we examine the fundamental connection between estimates of the linear predictor matrices obtained from unquantized data and those from quantized data via modified EDMD. Furthermore, using the law of large numbers, we demonstrate that, under a large data regime, the quantized estimate can be considered a regularized version of the unquantized estimate. We also explore the relationship between the two estimates in the finite data regime. We further analyze the effect of nonlinear lifting functions on this regularization due to quantization. The theory is validated through repeated numerical experiments conducted on several control systems. The effect of quantization on the MPC performance is also demonstrated.
This study aims to benchmark candidate strategies for embedding neural network (NN) surrogates in nonlinear model predictive control (NMPC) formulations that are subject to systems described with partial differential equations and that are solved via direct transcription (i.e., simultaneous methods). This study focuses on the use of physics-informed NNs and physics-informed convolutional NNs as the internal (surrogate) models within the NMPC formulation. One strategy embeds NN models as explicit algebraic constraints, leveraging the automatic differentiation (AD) of an algebraic modelling language (AML) to evaluate the derivatives. Alternatively, the solver can be provided with derivatives computed external to the AML via the AD routines of the machine learning environment the NN is trained in. The three numerical experiments considered in this work reveal that replacing mechanistic models with NN surrogates may not always offer computational advantages when smooth activation functions are used in conjunction with a local nonlinear solver (e.g., Ipopt), even with highly nonlinear systems. Moreover, in this context, the external function evaluation of the NN surrogates often outperforms the embedding strategies that rely on explicit algebraic constraints, likely due to the difficulty in initializing the auxiliary variables and constraints introduced by explicit algebraic reformulations.
This paper introduces a novel nonlinear model predictive control (NMPC) framework that incorporates a lifting technique to enhance control performance for nonlinear systems. While the lifting technique has been widely employed in linear systems to capture intersample behaviour, their application to nonlinear systems remains unexplored. We address this gap by formulating an NMPC scheme that combines fast-sample fast-hold (FSFH) approximations and numerical methods to approximate system dynamics and cost functions. The proposed approach is validated through two case studies: the Van der Pol oscillator and the inverted pendulum on a cart. Simulation results demonstrate that the lifted NMPC outperforms conventional NMPC in terms of reduced settling time and improved control accuracy. These findings underscore the potential of the lifting-based NMPC for efficient control of nonlinear systems, offering a practical solution for real-time applications.
Online motion planning is a challenging problem for intelligent robots moving in dense environments with dynamic obstacles, e.g., crowds. In this work, we propose a novel approach for optimal and safe online motion planning with minimal information about dynamic obstacles. Specifically, our approach requires only the current position of the obstacles and their maximum speed, but it does not need any information about their exact trajectories or dynamic model. The proposed methodology combines Monte Carlo Tree Search (MCTS), for online optimal planning via model simulations, with Velocity Obstacles (VO), for obstacle avoidance. We perform experiments in a cluttered simulated environment with walls, and up to 40 dynamic obstacles moving with random velocities and directions. With an ablation study, we show the key contribution of VO in scaling up the efficiency of MCTS, selecting the safest and most rewarding actions in the tree of simulations. Moreover, we show the superiority of our methodology with respect to state-of-the-art planners, including Non-linear Model Predictive Control (NMPC), in terms of improved collision rate, computational and task performance.
Digital Twin -- a virtual replica of a physical system enabling real-time monitoring, model updating, prediction, and decision-making -- combined with recent advances in machine learning, offers new opportunities for proactive control strategies in autonomous manufacturing. However, achieving real-time decision-making with Digital Twins requires efficient optimization driven by accurate predictions of highly nonlinear manufacturing systems. This paper presents a simultaneous multi-step Model Predictive Control (MPC) framework for real-time decision-making, using a multivariate deep neural network, named Time-Series Dense Encoder (TiDE), as the surrogate model. Unlike conventional MPC models which only provide one-step ahead prediction, TiDE is capable of predicting future states within the prediction horizon in one shot (multi-step), significantly accelerating the MPC. Using Directed Energy Deposition (DED) additive manufacturing as a case study, we demonstrate the effectiveness of the proposed MPC in achieving melt pool temperature tracking to ensure part quality, while reducing porosity defects by regulating laser power to maintain melt pool depth constraints. In this work, we first show that TiDE is capable of accurately predicting melt pool temperature and depth. Second, we demonstrate that the proposed MPC achieves precise temperature tracking while satisfying melt pool depth constraints within a targeted dilution range (10\%-30\%), reducing potential porosity defects. Compared to PID controller, the MPC results in smoother and less fluctuating laser power profiles with competitive or superior melt pool temperature control performance. This demonstrates the MPC's proactive control capabilities, leveraging time-series prediction and real-time optimization, positioning it as a powerful tool for future Digital Twin applications and real-time process optimization in manufacturing.
Functional Electrical Stimulation (FES) can be an effective tool to augment paretic muscle function and restore normal ankle function. Our approach incorporates a real-time, data-driven Model Predictive Control (MPC) scheme, built upon a Koopman operator theory (KOT) framework. This framework adeptly captures the complex nonlinear dynamics of ankle motion in a linearized form, enabling application of linear control approaches for highly nonlinear FES-actuated dynamics. Utilizing inertial measurement units (IMUs), our method accurately predicts the FES-induced ankle movements, while accounting for nonlinear muscle actuation dynamics, including the muscle activation for both plantarflexors, and dorsiflexors (Tibialis Anterior (TA)). The linear prediction model derived through KOT allowed us to formulate the MPC problem with linear state space dynamics, enhancing the real-time feasibility, precision and adaptability of the FES driven control. The effectiveness and applicability of our approach have been demonstrated through comprehensive simulations and experimental trials, including three participants with no disability and a participant with Multiple Sclerosis. Our findings highlight the potential of a KOT-based MPC approach for FES based gait assistance that offers effective and personalized assistance for individuals with gait impairment conditions.
This paper presents a framework for bounding the approximation error in imitation model predictive controllers utilizing neural networks. Leveraging the Lipschitz properties of these neural networks, we derive a bound that guides dataset design to ensure the approximation error remains at chosen limits. We discuss how this method can be used to design a stable neural network controller with performance guarantees employing existing robust model predictive control approaches for data generation. Additionally, we introduce a training adjustment, which is based on the sensitivities of the optimization problem and reduces dataset density requirements based on the derived bounds. We verify that the proposed augmentation results in improvements to the network's predictive capabilities and a reduction of the Lipschitz constant. Moreover, on a simulated inverted pendulum problem, we show that the approach results in a closer match of the closed-loop behavior between the imitation and the original model predictive controller.
Recent advancements in prompting techniques for Large Language Models (LLMs) have improved their reasoning, planning, and action abilities. This paper examines these prompting techniques through the lens of model predictive control (MPC). We show that LLMs act as implicit planning cost function minimizers when planning prompts are used. Under our framework we demonstrate that LLM planning performance can be improved further by incorporating real planning cost functions and evaluators.
This study presents autonomous guidance and control strategies for the purpose of reconfiguring close-range multi-satellite formations. The formation under consideration includes $N$ under-actuated deputy satellites and an uncontrolled virtual or physical chief spacecraft. The guidance problem is formulated as a trajectory optimization problem that incorporates typical dynamical and physical constraints, alongside a minimum acceleration threshold. This latter constraint arises from the physical limitations of the adopted low-thrust technology, which is commonly employed for precise, close-range relative orbital maneuvers. The guidance and control problem is addressed in two frameworks: centralized and distributed. The centralized approach provides a fuel-optimal solution, but it is practical only for formations with a small number of deputies. The distributed approach is more scalable but yields sub-optimal solutions. In the centralized framework, the chief is a physical satellite responsible for all calculations, while in the distributed framework, the chief is treated as a virtual point mass orbiting the Earth, and each deputy performs its own guidance and control calculations onboard. The study emphasizes the spaceborne implementation of the closed-loop control system, aiming for a reliable and automated solution to the optimal control problem. To this end, the risk of infeasibility is mitigated through first identifying the constraints that pose a potential threat of infeasibility, then properly softening them. Two Model Predictive Control architectures are implemented and compared, namely, a shrinking-horizon and a fixed-horizon schemes. Performances, in terms of fuel expenditure and achieved control accuracy, are analyzed on typical close-range reconfigurations requested by Earth observation missions and are compared against different implementations proposed in the literature.
In this paper, we present a novel cascade control structure with formal guarantees of uniform almost global asymptotic stability for the state tracking error dynamics of a quadcopter. The proposed approach features a model predictive control strategy for the outer loop, explicitly accounting for the non-zero total thrust constraint. The outer-loop controller generates an acceleration reference, which is then converted into attitude, angular velocity and acceleration references, subsequently tracked by a nonlinear inner-loop controller. The proposed cascade control strategy is validated through numerical case studies, underlying high-fidelity models, demonstrating its ability to track fast trajectories with small error.