Bart De Schutter

Minh Dang Doan, Pontus Giselsson, Tamás Keviczky et al.

A distributed model predictive control (DMPC) approach based on distributed optimization is applied to the power reference tracking problem of a hydro power valley (HPV) system. The applied optimization algorithm is based on accelerated gradient methods and achieves a convergence rate of O(1/k^2), where k is the iteration number. Major challenges in the control of the HPV include a nonlinear and large-scale model, nonsmoothness in the power-production functions, and a globally coupled cost function that prevents distributed schemes to be applied directly. We propose a linearization and approximation approach that accommodates the proposed the DMPC framework and provides very similar performance compared to a centralized solution in simulations. The provided numerical studies also suggest that for the sparsely interconnected system at hand, the distributed algorithm we propose is faster than a centralized state-of-the-art solver such as CPLEX.

Miguel Picallo Cruz, Adolfo Anta, Ara Panosyan et al.

In this work we study the problem of State Estimation(SE) in large-scale, 3-phase coupled, unbalanced distribution systems. More specifically, we address the problem of including mixed real-time measurements, synchronized and unsynchronized, from phasor measurement units and smart meters, into existing solutions. We propose a computationally efficient two-step method to update a prior solution using the measurements, while taking into account physical constraint caused by buses with no loads. We test the method on a benchmark test feeder to illustrate the effectiveness of the approach.

Minh Dang Doan, Tamás Keviczky, Bart De Schutter

We present a hierarchical model predictive control approach for large-scale systems based on dual decomposition. The proposed scheme allows coupling in both dynamics and constraints between the subsystems and generates a primal feasible solution within a finite number of iterations, using primal averaging and a constraint tightening approach. The primal update is performed in a distributed way and does not require exact solutions, while the dual problem uses an approximate subgradient method. Stability of the scheme is established using bounded suboptimality.

Miguel Picallo, Adolfo Anta, Bart De Schutter

The small amount of measurements in distribution grids makes their monitoring more difficult. Topological observability may not be possible, and thus, pseudo-measurements are needed to perform state estimation, which is required to control elements such as distributed generation or transformers at distribution grids. Therefore, we consider the problem of optimal sensor placement to improve the state estimation accuracy in large-scale, 3-phase coupled, unbalanced distribution grids. This is an NP-hard optimization problem whose optimal solution is unpractical to obtain for large networks. Therefore, we develop a computationally efficient convex optimization algorithm to compute a lower bound on the possible value of the optimal solution, and thus check the gap between the bound and heuristic solutions. We test the method on a large test feeder, the standard IEEE 8500-node, to show the effectiveness of the approach.

Laura Boca de Giuli, Samuel Mallick, Alessio La Bella et al.

This paper presents a model predictive control (MPC) framework leveraging an ensemble of data-based models to optimally control complex systems under multiple operating conditions. A novel combination rule for ensemble models is proposed, based on the statistical Mahalanobis distance, enabling the ensemble weights to suitably vary across the prediction window based on the system input. In addition, a novel state observer for ensemble models is developed using moving horizon estimation (MHE). The effectiveness of the proposed methodology is demonstrated on a benchmark energy system operating under multiple conditions.

Alessandro Riccardi, Thom Badings, Luca Laurenti et al.

The deployment of autonomous systems in safety-critical environments requires control policies that guarantee satisfaction of complex control specifications. These systems are commonly modeled as nonlinear discrete-time stochastic systems. A~popular approach to computing a policy that provably satisfies a complex control specification is to construct a finite-state abstraction, often represented as a Markov decision process (MDP) with intervals of transition probabilities, i.e., an interval MDP (IMDP). However, existing abstraction techniques compute a \emph{single policy}, thus leaving no room for online cost or performance optimization, e.g., of energy consumption. To overcome this limitation, we propose a novel IMDP abstraction technique that yields a \emph{set of policies}, each of which satisfies the control specification with a certain minimum probability. We can thus use any online control algorithm to search through this set of verified policies while retaining the guaranteed satisfaction probability of the entire policy set. In particular, we employ model predictive control (MPC) to minimize a desired cost function that is independent of the control specification considered in the abstraction. Our experiments demonstrate that our approach yields better control performance than state-of-the-art single-policy abstraction techniques, with a small degradation of the guarantees.

Anil Alan, Bart De Schutter

We study feasibility guarantees for safety filters developed using Control Barrier Functions (CBFs) when a safe set is defined using the pointwise minimum of continuously differentiable functions, a construction that is common for the backup CBF (BCBF) method and typically nonsmooth. We replace the minimum by its log-sum-exp (soft-min) smoothing and show that, under a strict safety condition, the smooth function becomes a CBF (or extended CBF) for a range of the smoothing parameter. For compact safe sets, we derive an explicit lower bound on the smoothing parameter that makes the smooth function a CBF and hence renders the corresponding safety constraint feasible. For unbounded sets, we introduce tail conditions under which the smooth function satisfies an extended CBF condition uniformly. Finally, we apply these results to BCBFs. We show that safety of a compact (terminal) backup set under a backup controller, together with a condition ensuring safety of the backup trajectories on the relevant boundary of the safe set, is sufficient for constraint feasibility for BCBFs. These results provide a recipe for a priori feasibility guarantees for smooth inner approximations of nonsmooth safe sets without the need for additional online certification.

Shengling Shi, Othmane Mazhar, Bart De Schutter

For the identification of switched systems with a measured switching signal, this work aims to analyze the effect of switching strategies on the estimation error. The data for identification is assumed to be collected from globally asymptotically or marginally stable switched systems under switches that are arbitrary or subject to an average dwell time constraint. Then the switched system is estimated by the least-squares (LS) estimator. To capture the effect of the parameters of the switching strategies on the LS estimation error, finite-sample error bounds are developed in this work. The obtained error bounds show that the estimation error is logarithmic of the switching parameters when there are only stable modes; however, when there are unstable modes, the estimation error bound can increase linearly as the switching parameter changes. This suggests that in the presence of unstable modes, the switching strategy should be properly designed to avoid the significant increase of the estimation error.

Xiaoyu Liu, Dimos V. Dimarogonas, Changxin Liu et al.

This paper presents a novel distributed model predictive control (MPC) formulation without terminal cost and a corresponding distributed synthesis approach for distributed linear discrete-time systems with coupled constraints. The proposed control scheme introduces an explicit stability condition as an additional constraint based on relaxed dynamic programming. As a result, contrary to other related approaches, system stability with the developed controller does not rely on designing a terminal cost. A distributed synthesis approach is then introduced to handle the stability constraint locally within each local agent. To solve the underlying optimization problem for distributed MPC, a violation-free distributed optimization approach is developed, using constraint tightening to ensure feasibility throughout iterations. A numerical example demonstrates that the proposed distributed MPC approach ensures closed-loop stability for each feasible control sequence, with each agent computing its control input in parallel.

Filippo Airaldi, Bart De Schutter, Azita Dabiri

In the backdrop of an increasingly pressing need for effective urban and highway transportation systems, this work explores the synergy between model-based and learning-based strategies to enhance traffic flow management by use of an innovative approach to the problem of ramp metering control that embeds Reinforcement Learning (RL) techniques within the Model Predictive Control (MPC) framework. The control problem is formulated as an RL task by crafting a suitable stage cost function that is representative of the traffic conditions, variability in the control action, and violations of the constraint on the maximum number of vehicles in queue. An MPC-based RL approach, which leverages the MPC optimal problem as a function approximation for the RL algorithm, is proposed to learn to efficiently control an on-ramp and satisfy its constraints despite uncertainties in the system model and variable demands. Simulations are performed on a benchmark small-scale highway network to compare the proposed methodology against other state-of-the-art control approaches. Results show that, starting from an MPC controller that has an imprecise model and is poorly tuned, the proposed methodology is able to effectively learn to improve the control policy such that congestion in the network is reduced and constraints are satisfied, yielding an improved performance that is superior to the other controllers.

Caio Fabio Oliveira da Silva, Azita Dabiri, Bart De Schutter

This work proposes an approach that integrates reinforcement learning and model predictive control (MPC) to solve finite-horizon optimal control problems in mixed-logical dynamical systems efficiently. Optimization-based control of such systems with discrete and continuous decision variables entails the online solution of mixed-integer linear programs, which suffer from the curse of dimensionality. Our approach aims to mitigate this issue by decoupling the decision on the discrete variables from the decision on the continuous variables. In the proposed approach, reinforcement learning determines the discrete decision variables and simplifies the online optimization problem of the MPC controller from a mixed-integer linear program to a linear program, significantly reducing the computational time. A fundamental contribution of this work is the definition of the decoupled Q-function, which plays a crucial role in making the learning problem tractable in a combinatorial action space. We motivate the use of recurrent neural networks to approximate the decoupled Q-function and show how they can be employed in a reinforcement learning setting. Simulation experiments on a microgrid system using real-world data demonstrate that the proposed method substantially reduces the online computation time of MPC while maintaining high feasibility and low suboptimality.

Archith Athrey, Othmane Mazhar, Meichen Guo et al.

In this paper, we analyze the regret incurred by a computationally efficient exploration strategy, known as naive exploration, for controlling unknown partially observable systems within the Linear Quadratic Gaussian (LQG) framework. We introduce a two-phase control algorithm called LQG-NAIVE, which involves an initial phase of injecting Gaussian input signals to obtain a system model, followed by a second phase of an interplay between naive exploration and control in an episodic fashion. We show that LQG-NAIVE achieves a regret growth rate of $\tilde{\mathcal{O}}(\sqrt{T})$, i.e., $\mathcal{O}(\sqrt{T})$ up to logarithmic factors after $T$ time steps, and we validate its performance through numerical simulations. Additionally, we propose LQG-IF2E, which extends the exploration signal to a `closed-loop' setting by incorporating the Fisher Information Matrix (FIM). We provide compelling numerical evidence of the competitive performance of LQG-IF2E compared to LQG-NAIVE.

Shengling Shi, Anastasios Tsiamis, Bart De Schutter

This work analyzes how the trade-off between the modeling error, the terminal value function error, and the prediction horizon affects the performance of a nominal receding-horizon linear quadratic (LQ) controller. By developing a novel perturbation result of the Riccati difference equation, a novel performance upper bound is obtained and suggests that for many cases, the prediction horizon can be either one or infinity to improve the control performance, depending on the relative difference between the modeling error and the terminal value function error. The result also shows that when an infinite horizon is desired, a finite prediction horizon that is larger than the controllability index can be sufficient for achieving a near-optimal performance, revealing a close relation between the prediction horizon and controllability. The obtained suboptimality performance upper bound is applied to provide novel sample complexity and regret guarantees for nominal receding-horizon LQ controllers in a learning-based setting. We show that an adaptive prediction horizon that increases as a logarithmic function of time is beneficial for regret minimization.

Samuel Mallick, Laura Boca de de Giuli, Alessio La Bella et al.

This paper addresses the design of an event-triggered, data-based, and performance-oriented adaption method for model predictive control (MPC). The performance of such a strategy strongly depends on the accuracy of the prediction model, which may require online adaption to prevent performance degradation under changing operating conditions. Unlike existing methods that continuously update model and control parameters from data, potentially leading to catastrophic forgetting and unnecessary control modifications, we propose a novel approach based on statistical monitoring of closed-loop performance indicators. This framework enables the detection of performance degradation, and, when required, controller adaption is performed via reinforcement learning and identification techniques. The proposed strategy is validated on a high-fidelity simulation of a district heating system benchmark.

R. R. Samani, A. Nunez, B. De Schutter

The growing volume of available infrastructural monitoring data enables the development of powerful datadriven approaches to estimate infrastructure health conditions using direct measurements. This paper proposes a deep learning methodology to estimate infrastructure physical parameters, such as railway track stiffness, using drive-by vibration response signals. The proposed method employs a Long Short-term Memory (LSTM) feature extractor accounting for temporal dependencies in the feature extraction phase, and a bidirectional Long Short-term Memory (BiLSTM) networks to leverage bidirectional temporal dependencies in both the forward and backward paths of the drive-by vibration response in condition estimation phase. Additionally, a framing approach is employed to enhance the resolution of the monitoring task to the beam level by segmenting the vibration signal into frames equal to the distance between individual beams, centering the frames over the beam nodes. The proposed LSTM-BiLSTM model offers a versatile tool for various bridge and railway infrastructure conditions monitoring using direct drive-by vibration response measurements. The results demonstrate the potential of incorporating temporal analysis in the feature extraction phase and emphasize the pivotal role of bidirectional temporal information in infrastructure health condition estimation. The proposed methodology can accurately and automatically estimate railway track stiffness and identify local stiffness reductions in the presence of noise using drive-by measurements. An illustrative case study of vehicle-track interaction simulation is used to demonstrate the performance of the proposed model, achieving a maximum mean absolute percentage error of 1.7% and 0.7% in estimating railpad and ballast stiffness, respectively.

Changrui Liu, Shengling Shi, Anil Alan et al.

Efficient energy management is essential for reliable and sustainable microgrid operation amid increasing renewable integration. This paper proposes an imitation learning-based framework to approximate mixed-integer Economic Model Predictive Control (EMPC) for microgrid energy management. The proposed method trains a neural network to imitate expert EMPC control actions from offline trajectories, enabling fast, real-time decision making without solving optimization problems online. To enhance robustness and generalization, the learning process includes noise injection during training to mitigate distribution shift and explicitly incorporates forecast uncertainty in renewable generation and demand. Simulation results demonstrate that the learned policy achieves economic performance comparable to EMPC while only requiring $10\%$ of the computation time of optimization-based EMPC in practice.

Reza Riahi Samani, Alfredo Nunez, Bart De Schutter

This paper presents a novel deep learning-based framework for infrastructure health monitoring using drive-by vibration response signals. Recognizing the importance of spectral and temporal information, we introduce the WaveletInception-BiLSTM network. The WaveletInception feature extractor utilizes a Learnable Wavelet Packet Transform (LWPT) as the stem for extracting vibration signal features, incorporating spectral information in the early network layers. This is followed by 1D Inception networks that extract multi-scale, high-level features at deeper layers. The extracted vibration signal features are then integrated with operational conditions via a Long Short-term Memory (LSTM) layer. The resulting feature extraction network effectively analyzes drive-by vibration signals across various measurement speeds without preprocessing and uses LSTM to capture interrelated temporal dependencies among different modes of information and to create feature vectors for health condition estimation. The estimator head is designed with a sequential modeling architecture using bidirectional LSTM (BiLSTM) networks, capturing bi-directional temporal relationships from drive-by measurements. This architecture allows for a high-resolution, beam-level assessment of infrastructure health conditions. A case study focusing on railway track stiffness estimation with simulated drive-by vibration signals shows that the model significantly outperforms state-of-the-art methods in estimating railway ballast and railpad stiffness parameters. Results underscore the potential of this approach for accurate, localized, and fully automated drive-by infrastructure health monitoring.

Filippo Airaldi, Bart De Schutter, Azita Dabiri

Unconstrained global optimisation aims to optimise expensive-to-evaluate black-box functions without gradient information. Bayesian optimisation, one of the most well-known techniques, typically employs Gaussian processes as surrogate models, leveraging their probabilistic nature to balance exploration and exploitation. However, Gaussian processes become computationally prohibitive in high-dimensional spaces. Recent alternatives, based on inverse distance weighting (IDW) and radial basis functions (RBFs), offer competitive, computationally lighter solutions. Despite their efficiency, both traditional global and Bayesian optimisation strategies suffer from the myopic nature of their acquisition functions, which focus solely on immediate improvement neglecting future implications of the sequential decision making process. Nonmyopic acquisition functions devised for the Bayesian setting have shown promise in improving long-term performance. Yet, their use in deterministic strategies with IDW and RBF remains unexplored. In this work, we introduce novel nonmyopic acquisition strategies tailored to IDW- and RBF-based global optimisation. Specifically, we develop dynamic programming-based paradigms, including rollout and multi-step scenario-based optimisation schemes, to enable lookahead acquisition. These methods optimise a sequence of query points over a horizon (instead of only at the next step) by predicting the evolution of the surrogate model, inherently managing the exploration-exploitation trade-off in a systematic way via optimisation techniques. The proposed approach represents a significant advance in extending nonmyopic acquisition principles, previously confined to Bayesian optimisation, to the deterministic framework. Empirical results on synthetic and hyperparameter tuning benchmark problems demonstrate that these nonmyopic methods outperform conventional myopic approaches.

Erwin de Gelder, Jasper Hof, Eric Cator et al.

The development of assessment methods for the performance of Automated Vehicles (AVs) is essential to enable the deployment of automated driving technologies, due to the complex operational domain of AVs. One candidate is scenario-based assessment, in which test cases are derived from real-world road traffic scenarios obtained from driving data. Because of the high variety of the possible scenarios, using only observed scenarios for the assessment is not sufficient. Therefore, methods for generating additional scenarios are necessary. Our contribution is twofold. First, we propose a method to determine the parameters that describe the scenarios to a sufficient degree without relying on strong assumptions on the parameters that characterize the scenarios. By estimating the probability density function (pdf) of these parameters, realistic parameter values can be generated. Second, we present the Scenario Representativeness (SR) metric based on the Wasserstein distance, which quantifies to what extent the scenarios with the generated parameter values are representative of real-world scenarios while covering the actual variety found in the real-world scenarios. A comparison of our proposed method with methods relying on assumptions of the scenario parametrization and pdf estimation shows that the proposed method can automatically determine the optimal scenario parametrization and pdf estimation. Furthermore, we demonstrate that our SR metric can be used to choose the (number of) parameters that best describe a scenario. The presented method is promising, because the parameterization and pdf estimation can directly be applied to already available importance sampling strategies for accelerating the evaluation of AVs.

Erwin de Gelder, Eric Cator, Jan-Pieter Paardekooper et al.

The safety assessment of automated vehicles (AVs) is an important aspect of the development cycle of AVs. A scenario-based assessment approach is accepted by many players in the field as part of the complete safety assessment. A scenario is a representation of a situation on the road to which the AV needs to respond appropriately. One way to generate the required scenario-based test descriptions is to parameterize the scenarios and to draw these parameters from a probability density function (pdf). Because the shape of the pdf is unknown beforehand, assuming a functional form of the pdf and fitting the parameters to the data may lead to inaccurate fits. As an alternative, Kernel Density Estimation (KDE) is a promising candidate for estimating the underlying pdf, because it is flexible with the underlying distribution of the parameters. Drawing random samples from a pdf estimated with KDE is possible without the need of evaluating the actual pdf, which makes it suitable for drawing random samples for, e.g., Monte Carlo methods. Sampling from a KDE while the samples satisfy a linear equality constraint, however, has not been described in the literature, as far as the authors know. In this paper, we propose a method to sample from a pdf estimated using KDE, such that the samples satisfy a linear equality constraint. We also present an algorithm of our method in pseudo-code. The method can be used to generating scenarios that have, e.g., a predetermined starting speed or to generate different types of scenarios. This paper also shows that the method for sampling scenarios can be used in case a Singular Value Decomposition (SVD) is used to reduce the dimension of the parameter vectors.

Jesus Lago, Grzegorz Marcjasz, Bart De Schutter et al.

While the field of electricity price forecasting has benefited from plenty of contributions in the last two decades, it arguably lacks a rigorous approach to evaluating new predictive algorithms. The latter are often compared using unique, not publicly available datasets and across too short and limited to one market test samples. The proposed new methods are rarely benchmarked against well established and well performing simpler models, the accuracy metrics are sometimes inadequate and testing the significance of differences in predictive performance is seldom conducted. Consequently, it is not clear which methods perform well nor what are the best practices when forecasting electricity prices. In this paper, we tackle these issues by performing a literature survey of state-of-the-art models, comparing state-of-the-art statistical and deep learning methods across multiple years and markets, and by putting forward a set of best practices. In addition, we make available the considered datasets, forecasts of the state-of-the-art models, and a specifically designed python toolbox, so that new algorithms can be rigorously evaluated in future studies.

E. de Gelder, J. -P. Paardekooper, A. Khabbaz Saberi et al.

The development of new assessment methods for the performance of automated vehicles is essential to enable the deployment of automated driving technologies, due to the complex operational domain of automated vehicles. One contributing method is scenario-based assessment in which test cases are derived from real-world road traffic scenarios obtained from driving data. Given the complexity of the reality that is being modeled in these scenarios, it is a challenge to define a structure for capturing these scenarios. An intensional definition that provides a set of characteristics that are deemed to be both necessary and sufficient to qualify as a scenario assures that the scenarios constructed are both complete and intercomparable. In this article, we develop a comprehensive and operable definition of the notion of scenario while considering existing definitions in the literature. This is achieved by proposing an object-oriented framework in which scenarios and their building blocks are defined as classes of objects having attributes, methods, and relationships with other objects. The object-oriented approach promotes clarity, modularity, reusability, and encapsulation of the objects. We provide definitions and justifications of each of the terms. Furthermore, the framework is used to translate the terms in a coding language that is publicly available.

Jesus Lago, Karel De Brabandere, Fjo De Ridder et al.

Due to the increasing integration of solar power into the electrical grid, forecasting short-term solar irradiance has become key for many applications, e.g.~operational planning, power purchases, reserve activation, etc. In this context, as solar generators are geographically dispersed and ground measurements are not always easy to obtain, it is very important to have general models that can predict solar irradiance without the need of local data. In this paper, a model that can perform short-term forecasting of solar irradiance in any general location without the need of ground measurements is proposed. To do so, the model considers satellite-based measurements and weather-based forecasts, and employs a deep neural network structure that is able to generalize across locations; particularly, the network is trained only using a small subset of sites where ground data is available, and the model is able to generalize to a much larger number of locations where ground data does not exist. As a case study, 25 locations in The Netherlands are considered and the proposed model is compared against four local models that are individually trained for each location using ground measurements. Despite the general nature of the model, it is shown show that the proposed model is equal or better than the local models: when comparing the average performance across all the locations and prediction horizons, the proposed model obtains a 31.31% rRMSE (relative root mean square error) while the best local model achieves a 32.01% rRMSE.

Jesus Lago, Fjo De Ridder, Peter Vrancx et al.

Motivated by the increasing integration among electricity markets, in this paper we propose two different methods to incorporate market integration in electricity price forecasting and to improve the predictive performance. First, we propose a deep neural network that considers features from connected markets to improve the predictive accuracy in a local market. To measure the importance of these features, we propose a novel feature selection algorithm that, by using Bayesian optimization and functional analysis of variance, evaluates the effect of the features on the algorithm performance. In addition, using market integration, we propose a second model that, by simultaneously predicting prices from two markets, improves the forecasting accuracy even further. As a case study, we consider the electricity market in Belgium and the improvements in forecasting accuracy when using various French electricity features. We show that the two proposed models lead to improvements that are statistically significant. Particularly, due to market integration, the predictive accuracy is improved from 15.7% to 12.5% sMAPE (symmetric mean absolute percentage error). In addition, we show that the proposed feature selection algorithm is able to perform a correct assessment, i.e. to discard the irrelevant features.

Frederik Ruelens, Bert Claessens, Salman Quaiyum et al.

Electric water heaters have the ability to store energy in their water buffer without impacting the comfort of the end user. This feature makes them a prime candidate for residential demand response. However, the stochastic and nonlinear dynamics of electric water heaters, makes it challenging to harness their flexibility. Driven by this challenge, this paper formulates the underlying sequential decision-making problem as a Markov decision process and uses techniques from reinforcement learning. Specifically, we apply an auto-encoder network to find a compact feature representation of the sensor measurements, which helps to mitigate the curse of dimensionality. A wellknown batch reinforcement learning technique, fitted Q-iteration, is used to find a control policy, given this feature representation. In a simulation-based experiment using an electric water heater with 50 temperature sensors, the proposed method was able to achieve good policies much faster than when using the full state information. In a lab experiment, we apply fitted Q-iteration to an electric water heater with eight temperature sensors. Further reducing the state vector did not improve the results of fitted Q-iteration. The results of the lab experiment, spanning 40 days, indicate that compared to a thermostat controller, the presented approach was able to reduce the total cost of energy consumption of the electric water heater by 15%.

Frederik Ruelens, Bert Claessens, Stijn Vandael et al.

Driven by recent advances in batch Reinforcement Learning (RL), this paper contributes to the application of batch RL to demand response. In contrast to conventional model-based approaches, batch RL techniques do not require a system identification step, which makes them more suitable for a large-scale implementation. This paper extends fitted Q-iteration, a standard batch RL technique, to the situation where a forecast of the exogenous data is provided. In general, batch RL techniques do not rely on expert knowledge on the system dynamics or the solution. However, if some expert knowledge is provided, it can be incorporated by using our novel policy adjustment method. Finally, we tackle the challenge of finding an open-loop schedule required to participate in the day-ahead market. We propose a model-free Monte-Carlo estimator method that uses a metric to construct artificial trajectories and we illustrate this method by finding the day-ahead schedule of a heat-pump thermostat. Our experiments show that batch RL techniques provide a valuable alternative to model-based controllers and that they can be used to construct both closed-loop and open-loop policies.