This paper studies uncertainty set estimation for unknown linear systems. Uncertainty sets are crucial for the quality of robust control since they directly influence the conservativeness of the control design. Departing from the confidence region analysis of least squares estimation, this paper focuses on set membership estimation (SME). Though good numerical performances have attracted applications of SME in the control literature, the non-asymptotic convergence rate of SME for linear systems remains an open question. This paper provides the first convergence rate bounds for SME and discusses variations of SME under relaxed assumptions. We also provide numerical results demonstrating SME's practical promise.
We analyse the conservatism and regret of distributionally robust (DR) stochastic model predictive control (SMPC) when using moment-based ambiguity sets for modeling unknown uncertainties. To quantify the conservatism, we compare the deterministic constraint tightening while taking a DR approach against the optimal tightening when the exact distributions of the stochastic uncertainties are known. Furthermore, we quantify the regret by comparing the performance when the distributions of the stochastic uncertainties are known and unknown. Analysing the accumulated sub-optimality of SMPC due to the lack of knowledge about the true distributions of the uncertainties marks the novel contribution of this work.
In this paper we propose a framework towards achieving two intertwined objectives: (i) equipping reinforcement learning with active exploration and deliberate information gathering, such that it regulates state and parameter uncertainties resulting from modeling mismatches and noisy sensory; and (ii) overcoming the computational intractability of stochastic optimal control. We approach both objectives by using reinforcement learning to compute the stochastic optimal control law. On one hand, we avoid the curse of dimensionality prohibiting the direct solution of the stochastic dynamic programming equation. On the other hand, the resulting stochastic optimal control reinforcement learning agent admits caution and probing, that is, optimal online exploration and exploitation. Unlike fixed exploration and exploitation balance, caution and probing are employed automatically by the controller in real-time, even after the learning process is terminated. We conclude the paper with a numerical simulation, illustrating how a Linear Quadratic Regulator with the certainty equivalence assumption may lead to poor performance and filter divergence, while our proposed approach is stabilizing, of an acceptable performance, and computationally convenient.
Learning-based controllers have demonstrated superior performance compared to classical controllers in various tasks. However, providing safety guarantees is not trivial. Safety, the satisfaction of state and input constraints, can be guaranteed by augmenting the learned control policy with a safety filter. Model predictive safety filters (MPSFs) are a common safety filtering approach based on model predictive control (MPC). MPSFs seek to guarantee safety while minimizing the difference between the proposed and applied inputs in the immediate next time step. This limited foresight can lead to jerky motions and undesired oscillations close to constraint boundaries, known as chattering. In this paper, we reduce chattering by considering input corrections over a longer horizon. Under the assumption of bounded model uncertainties, we prove recursive feasibility using techniques from robust MPC. We verified the proposed approach in both extensive simulation and quadrotor experiments. In experiments with a Crazyflie 2.0 drone, we show that, in addition to preserving the desired safety guarantees, the proposed MPSF reduces chattering by more than a factor of 4 compared to previous MPSF formulations.
We present Ungar, an open-source library to aid the implementation of high-dimensional optimal control problems (OCPs). We adopt modern template metaprogramming techniques to enable the compile-time modeling of complex systems while retaining maximum runtime efficiency. Our framework provides syntactic sugar to allow for expressive formulations of a rich set of structured dynamical systems. While the core modules depend only on the header-only Eigen and Boost.Hana libraries, we bundle our codebase with optional packages and custom wrappers for automatic differentiation, code generation, and nonlinear programming. Finally, we demonstrate the versatility of Ungar in various model predictive control applications, namely, four-legged locomotion and collaborative loco-manipulation with multiple one-armed quadruped robots. Ungar is available under the Apache License 2.0 at https://github.com/fdevinc/ungar.
Due to the nonlinearities and operational constraints typical to quadcopter missions, Model Predictive Control (MPC) encounters the major challenge of high computational power necessary for the online implementation. This problem may prove impractical, especially for a hardware-limited or small-scale setup. By removing the need for online solvers while keeping the constraint satisfaction and optimality, Explicit MPC (ExMPC) stands out as a strong candidate for this application. Yet, the formulation was usually hindered by the two main problems: nonlinearity and dimensionality. In this paper, we propose an ExMPC solution for the quadcopter position stabilization by analyzing its description (dynamics and constraints) in the flat output space. With the former issue, the system is exactly linearized into a concatenation of three double integrators at a price of cumbersome constraints in the new coordinates. For the latter, with a suitable characterization of these constraints, the stabilizing ExMPC can be computed for each double integrator separately. The controller is then validated via simulations and experimental tests. The proposed scheme achieves similar performance and guarantees to the state-of-the-art solution but with notably less computational effort, allowing scalability in a centralized manner
A trust-aware safe control system for autonomous navigation in the presence of humans, specifically pedestrians, is presented. The system combines model predictive control (MPC) with control barrier functions (CBFs) and trust estimation to ensure safe and reliable navigation in complex environments. Pedestrian trust values are computed based on features, extracted from camera sensor images, such as mutual eye contact and smartphone usage. These trust values are integrated into the MPC controller's CBF constraints, allowing the autonomous vehicle to make informed decisions considering pedestrian behavior. Simulations conducted in the CARLA driving simulator demonstrate the feasibility and effectiveness of the proposed system, showcasing more conservative behaviour around inattentive pedestrians and vice versa. The results highlight the practicality of the system in real-world applications, providing a promising approach to enhance the safety and reliability of autonomous navigation systems, especially self-driving vehicles.
Learning-based optimal control algorithms control unknown systems using past trajectory data and a learned model of the system dynamics. These controllers use either a linear approximation of the learned dynamics, trading performance for faster computation, or nonlinear optimization methods, which typically perform better but can limit real-time applicability. In this work, we present a novel nonlinear controller that exploits differential flatness to achieve similar performance to state-of-the-art learning-based controllers but with significantly less computational effort. Differential flatness is a property of dynamical systems whereby nonlinear systems can be exactly linearized through a nonlinear input mapping. Here, the nonlinear transformation is learned as a Gaussian process and is used in a safety filter that guarantees, with high probability, stability as well as input and flat state constraint satisfaction. This safety filter is then used to refine inputs from a flat model predictive controller to perform constrained nonlinear learning-based optimal control through two successive convex optimizations. We compare our method to state-of-the-art learning-based control strategies and achieve similar performance, but with significantly better computational efficiency, while also respecting flat state and input constraints, and guaranteeing stability.
This paper studies the design of a Model Predictive Controller (MPC) for integrated lateral stability, traction/braking control, and rollover prevention of electric vehicles intended for very high speed (VHS) racing applications. We first identify the advantages of a state-of-the-art dynamic model in that it includes rollover prevention into the MPC (a total of 8 states) and also linearizes the tire model prior to solving the MPC problem to save computation time. Then the design of a novel model predictive controller for lateral stability control is proposed aimed for achieving stable control at top speed significantly greater than typical highway speed limits. We have tested the new solution in simulation environments associated with the Indy Autonomous Challenge, where its real-world racing conditions include significant road banking angles, lateral position tracking, and a different suspension model of its Dallara Indy Lights chassis. The results are very promising with a low solver time in Python, as low as 50 Hz, and a lateral error of 30 cm at speeds of 45 m/s. Our open source code is available at: https: //github.com/jadyahya/Roll-Yaw-and-Lateral-Velocity-MPC/.
This paper studies a heterogeneous Unmanned Aerial Vehicles (UAVs) cooperative search approach suitable for complex environments. In the application, a fixed-wing UAV drops rotor UAVs to deploy the cluster rapidly. Meanwhile, the fixed-wing UAV works as a communication relay node to improve the search performance of the cluster further. The distributed model predictive control and genetic algorithms are adopted to make online intelligent decisions on UAVs search directions. On this basis, a jump grid decision method is proposed to satisfy the maneuverability constraints of UAVs, a parameter dynamic selection method is developed to make search decisions more responsive to task requirements, and a search information transmission method with low bandwidth is designed. This approach can enable UAVs to discover targets quickly, cope with various constraints and unexpected situations, and make adaptive decisions, significantly improving the robustness of search tasks in complex, dynamic, and unknown environments. The proposed approach is tested with several search scenarios, and simulation results show that the cooperative search performance of heterogeneous UAVs is significantly improved compared to homogeneous UAVs.
Genetic algorithm (GA) is typically used to solve nonlinear model predictive control's optimization problem. However, the size of the search space in which the GA searches for the optimal control inputs is crucial for its applicability to fast-response systems. This paper proposes accelerating the genetic optimization of NMPC by learning optimal search space size. The approach trains a multivariate regression model to adaptively predict the best smallest size of the search space in every control cycle. The proposed approach reduces the GA's computational time, improves the chance of convergence to better control inputs, and provides a stable and feasible solution. The proposed approach was evaluated on three nonlinear systems and compared to four other evolutionary algorithms implemented in a processor-in-the-loop fashion. The results show that the proposed approach provides a 17-45\% reduction in computational time and increases the convergence rate by 35-47\%. The source code is available on GitHub.
In this paper, an offset-free bilinear model predictive control approach for wind turbines is presented. State-of-the-art controllers employ different control loops for pitch angle and generator torque which switch depending on wind conditions. In contrast, the presented controller is based on one unified control law that works for all wind conditions. The inherent nonlinearity of wind turbines is addressed through a piecewise-affine approximation of the power coefficient, which is modelled in a mixed-integer fashion. The presented controller is compared to a state-of-the-art baseline controller in a numerical case study using OpenFAST. Simulation results show that the presented controller ensures accurate reference power tracking. Additionally, damage equivalent loads are reduced for higher wind speeds.
Model predictive control (MPC) achieves stability and constraint satisfaction for general nonlinear systems, but requires computationally expensive online optimization. This paper studies approximations of such MPC controllers via neural networks (NNs) to achieve fast online evaluation. We propose safety augmentation that yields deterministic guarantees for convergence and constraint satisfaction despite approximation inaccuracies. We approximate the entire input sequence of the MPC with NNs, which allows us to verify online if it is a feasible solution to the MPC problem. We replace the NN solution by a safe candidate based on standard MPC techniques whenever it is infeasible or has worse cost. Our method requires a single evaluation of the NN and forward integration of the input sequence online, which is fast to compute on resource-constrained systems. The proposed control framework is illustrated using two numerical non-linear MPC benchmarks of different complexity, demonstrating computational speedups that are orders of magnitude higher than online optimization. In the examples, we achieve deterministic safety through the safety-augmented NNs, where a naive NN implementation fails.
Energy management systems (EMS) have traditionally been implemented using rule-based control (RBC) and model predictive control (MPC) methods. However, recent research has explored the use of reinforcement learning (RL) as a promising alternative. This paper introduces TreeC, a machine learning method that utilizes the covariance matrix adaptation evolution strategy metaheuristic algorithm to generate an interpretable EMS modeled as a decision tree. Unlike RBC and MPC approaches, TreeC learns the decision strategy of the EMS based on historical data, adapting the control model to the controlled energy grid. The decision strategy is represented as a decision tree, providing interpretability compared to RL methods that often rely on black-box models like neural networks. TreeC is evaluated against MPC with perfect forecast and RL EMSs in two case studies taken from literature: an electric grid case and a household heating case. In the electric grid case, TreeC achieves an average energy loss and constraint violation score of 19.2, which is close to MPC and RL EMSs that achieve scores of 14.4 and 16.2 respectively. All three methods control the electric grid well especially when compared to the random EMS, which obtains an average score of 12 875. In the household heating case, TreeC performs similarly to MPC on the adjusted and averaged electricity cost and total discomfort (0.033 EUR/m$^2$ and 0.42 Kh for TreeC compared to 0.037 EUR/m$^2$ and 2.91 kH for MPC), while outperforming RL (0.266 EUR/m$^2$ and 24.41 Kh).
Over the last few years, we have not seen any major developments in model-free or model-based learning methods that would make one obsolete relative to the other. In most cases, the used technique is heavily dependent on the use case scenario or other attributes, e.g. the environment. Both approaches have their own advantages, for example, sample efficiency or computational efficiency. However, when combining the two, the advantages of each can be combined and hence achieve better performance. The TD-MPC framework is an example of this approach. On the one hand, a world model in combination with model predictive control is used to get a good initial estimate of the value function. On the other hand, a Q function is used to provide a good long-term estimate. Similar to algorithms like MuZero a latent state representation is used, where only task-relevant information is encoded to reduce the complexity. In this paper, we propose the use of a reconstruction function within the TD-MPC framework, so that the agent can reconstruct the original observation given the internal state representation. This allows our agent to have a more stable learning signal during training and also improves sample efficiency. Our proposed addition of another loss term leads to improved performance on both state- and image-based tasks from the DeepMind-Control suite.
We study the convex hulls of reachable sets of nonlinear systems with bounded disturbances and uncertain initial conditions. Reachable sets play a critical role in control, but remain notoriously challenging to compute, and existing over-approximation tools tend to be conservative or computationally expensive. In this work, we characterize the convex hulls of reachable sets as the convex hulls of solutions of an ordinary differential equation with initial conditions on the sphere. This finite-dimensional characterization unlocks an efficient sampling-based estimation algorithm to accurately over-approximate reachable sets. We also study the structure of the boundary of the reachable convex hulls and derive error bounds for the estimation algorithm. We give applications to neural feedback loop analysis and robust MPC.
We propose a supervised learning framework for computing solutions of multi-parametric Mixed Integer Linear Programs (MILPs) that arise in Model Predictive Control. Our approach also quantifies sub-optimality for the computed solutions. Inspired by Branch-and-Bound techniques, the key idea is to train a Neural Network/Random Forest, which for a given parameter, predicts a strategy consisting of (1) a set of Linear Programs (LPs) such that their feasible sets form a partition of the feasible set of the MILP and (2) a candidate integer solution. For control computation and sub-optimality quantification, we solve a set of LPs online in parallel. We demonstrate our approach for a motion planning example and compare against various commercial and open-source mixed-integer programming solvers.
We propose an iterative approach for designing Robust Learning Model Predictive Control (LMPC) policies for a class of nonlinear systems with additive, unmodelled dynamics. The nominal dynamics are assumed to be difference flat, i.e., the state and input can be reconstructed using flat output sequences. For the considered class of systems, we synthesize Robust MPC policies and show how to use historical trajectory data collected during iterative tasks to 1) obtain bounds on the unmodelled dynamics and 2) construct a convex value function approximation along with a convex safe set in the space of output sequences for designing terminal components in the Robust MPC design. We show that the proposed strategy guarantees robust constraint satisfaction, asymptotic convergence to a desired subset of the state space, and non-decreasing closed-loop performance at each policy update. Finally, simulation results demonstrate the effectiveness of the proposed strategy on a minimum time control problem using a constrained nonlinear and uncertain vehicle model.
Recent advances in model predictive control (MPC) leverage local communication constraints to produce localized MPC algorithms whose complexities scale independently of total network size. However, no characterization is available regarding global performance, i.e. whether localized MPC (with communication constraints) performs just as well as global MPC (no communication constraints). In this paper, we provide analysis and guarantees on global performance of localized MPC -- in particular, we derive sufficient conditions for optimal global performance in the presence of local communication constraints. We also present an algorithm to determine the communication structure for a given system that will preserve performance while minimizing computational complexity. The effectiveness of the algorithm is verified in simulations, and additional relationships between network properties and performance-preserving communication constraints are characterized. A striking finding is that in a network of 121 coupled pendula, each node only needs to communicate with its immediate neighbors to preserve optimal global performance. Overall, this work offers theoretical understanding on the effect of local communication on global performance, and provides practitioners with the tools necessary to deploy localized model predictive control by establishing a rigorous method of selecting local communication constraints. This work also demonstrates -- surprisingly -- that the inclusion of severe communication constraints need not compromise global performance.
In this paper, we propose a novel approach for computing robust backward reachable sets from noisy data for unknown constrained linear systems subject to bounded disturbances. In particular, we develop an algorithm for obtaining zonotopic inner approximations that can be used for control purposes. It is shown that such sets, if built on an extended space including states and inputs, can be used to embed the system's one-step evolution in the computed extended regions. Such a result is then exploited to build a set-theoretic model predictive controller that, offline, builds a recursive family of robust data-driven reachable sets, and online, computes recursively admissible control actions without explicitly resorting to either a model of the system or the available data.The validity of the proposed data-driven solution is verified by means of a numerical simulation and its performance is contrasted with the model-based counterpart.