This paper studies the problem of distributionally robust model predictive control (MPC) using total variation distance ambiguity sets. For a discrete-time linear system with additive disturbances, we provide a conditional value-at-risk reformulation of the MPC optimization problem that is distributionally robust in the expected cost and chance constraints. The distributionally robust chance constraint is over-approximated as a simpler, tightened chance constraint that reduces the computational burden. Numerical experiments support our results on probabilistic guarantees and computational efficiency.
We propose a robust model predictive control (MPC) method for discrete-time linear systems with polytopic model uncertainty and additive disturbances. Optimizing over linear time-varying (LTV) state feedback controllers has been successfully used for robust MPC when only additive disturbances are present. However, it is challenging to design LTV state feedback controllers in the face of model uncertainty whose effects are difficult to bound. To address this issue, we propose a novel approach to over-approximate the effects of both model uncertainty and additive disturbances by a filtered additive disturbance signal. Using the System Level Synthesis framework, we jointly search for robust LTV state feedback controllers and the bounds on the effects of uncertainty online, which allows us to reduce the conservatism and minimize an upper bound on the worst-case cost in robust MPC. We provide a comprehensive numerical comparison of our method and representative robust MPC methods from the literature. Numerical examples demonstrate that our proposed method can significantly reduce the conservatism over a wide range of uncertainty parameters with comparable computational effort as the baseline methods.
For model-based control, an accurate and in its complexity suitable representation of the real system is a decisive prerequisite for high and robust control quality. In a structured step-by-step procedure, a model predictive control (MPC) scheme for a Schunk PowerCube robot is derived. Neweul-M$^2$ provides the necessary nonlinear model in symbolical and numerical form. To handle the heavy online computational burden involved with the derived nonlinear model, a linear time-varying MPC scheme is developed based on linearizing the nonlinear system concerning the desired trajectory and the a priori known corresponding feed-forward controller. To improve the identification of the nonlinear friction models of the joints, a nonlinear regression method and the Sparse Identification of Nonlinear Dynamics (SINDy) are compared with each other concerning robustness, online adaptivity, and necessary preprocessing of the input data. Everything is implemented on a slim, low-cost control system with a standard laptop PC.
Data-driven predictive control (DDPC) has been recently proposed as an effective alternative to traditional model-predictive control (MPC) for its unique features of being time-efficient and unbiased with respect to the oracle solution. Nonetheless, it has also been observed that noise may strongly jeopardize the final closed-loop performance since it affects both the data-based system representation and the control update computed from the online measurements. Recent studies have shown that regularization is potentially a successful tool to counteract the effect of noise. At the same time, regularization requires the tuning of a set of penalty terms, whose choice might be practically difficult without closed-loop experiments. In this paper, by means of subspace identification tools, we pursue a three-fold goal: $(i)$ we set up a unified framework for the existing regularized data-driven predictive control schemes for stochastic systems; $(ii)$ we introduce $\gamma$-DDPC, an efficient two-stage scheme that splits the optimization problem into two parts: fitting the initial conditions and optimizing the future performance, while guaranteeing constraint satisfaction; $(iii)$ we discuss the role of regularization for data-driven predictive control, providing new insight on $when$ and $how$ it should be applied. A benchmark numerical case study finally illustrates the performance of $\gamma$-DDPC, showing how controller design can be simplified in terms of tuning effort and computational complexity when benefiting from the insights coming from the subspace identification realm.
In this paper, we present a systematic approach for high-performance and efficient trajectory tracking control of autonomous wheel loaders. With the nonlinear dynamic model of a wheel loader, nonlinear model predictive control (MPC) is used in offline trajectory planning to obtain a high-performance state-control trajectory while satisfying the state and control constraints. In tracking control, the nonlinear model is embedded into a Linear Parameter Varying (LPV) model and the LPV-MPC strategy is used to achieve fast online computation and good tracking performance. To demonstrate the effectiveness and the advantages of the LPV-MPC, we test and compare three model predictive control strategies in the high-fidelity simulation environment. With the planned trajectory, three tracking control strategies LPV-MPC, nonlinear MPC, and LTI-MPC are simulated and compared in the perspectives of computational burden and tracking performance. The LPV-MPC can achieve better performance than conventional LTI-MPC because more accurate nominal system dynamics are captured in the LPV model. In addition, LPV-MPC achieves slightly worse tracking performance but tremendously improved computational efficiency than nonlinear MPC. A video with loading cycles completed by our autonomous wheel loader in the simulation environment can be found here: https://youtu.be/QbNfS_wZKKA.
In this work, we propose a distributionally robust stochastic model predictive control (DR-SMPC) algorithm to address the problem of two-sided chance constrained discrete-time linear system corrupted by additive noise. The prevalent mechanism to cope with two-sided chance constraints is the so-called risk allocation approach, which conservatively approximates the two-sided chance constraints with two single chance constraints by applying the Boole's inequality. In this proposed DR-SMPC framework, an exact tractable second-order cone (SOC) approach is adopted to abstract the two-sided chance constraints by considering the first and second moments of the noise. The proposed DR-SMPC algorithm is able to guarantee that the worst-case probability of violating both the upper and lower limits of safety constraints is within the pre-specified maximum probability (PsMP). By flexibly adjusting this PsMP, the feasible region of the initial states can be increased for the SMPC problem. The recursive feasibility and convergence of the proposed DR-SMPC are established rigorously by introducing binary initialization strategy of nominal state. Simulation studies of two practical cases are conducted to demonstrate the effectiveness of the proposed DR-SMPC algorithm.
Model Predictive Control (MPC) has become a popular framework in embedded control for high-performance autonomous systems. However, to achieve good control performance using MPC, an accurate dynamics model is key. To maintain real-time operation, the dynamics models used on embedded systems have been limited to simple first-principle models, which substantially limits their representative power. In contrast to such simple models, machine learning approaches, specifically neural networks, have been shown to accurately model even complex dynamic effects, but their large computational complexity hindered combination with fast real-time iteration loops. With this work, we present Real-time Neural MPC, a framework to efficiently integrate large, complex neural network architectures as dynamics models within a model-predictive control pipeline. Our experiments, performed in simulation and the real world onboard a highly agile quadrotor platform, demonstrate the capabilities of the described system to run learned models with, previously infeasible, large modeling capacity using gradient-based online optimization MPC. Compared to prior implementations of neural networks in online optimization MPC we can leverage models of over 4000 times larger parametric capacity in a 50Hz real-time window on an embedded platform. Further, we show the feasibility of our framework on real-world problems by reducing the positional tracking error by up to 82% when compared to state-of-the-art MPC approaches without neural network dynamics.
District cooling energy plants (DCEPs) consisting of chillers, cooling towers, and thermal energy storage (TES) systems consume a considerable amount of electricity. Optimizing the scheduling of the TES and chillers to take advantage of time-varying electricity price is a challenging optimal control problem. The classical method, model predictive control (MPC), requires solving a high dimensional mixed-integer nonlinear program (MINLP) because of the on/off actuation of the chillers and charging/discharging of TES, which are computationally challenging. RL is an attractive alternative to MPC: the real time control computation is a low-dimensional optimization problem that can be easily solved. However, the performance of an RL controller depends on many design choices. In this paper, we propose a Q-learning based reinforcement learning (RL) controller for this problem. Numerical simulation results show that the proposed RL controller is able to reduce energy cost over a rule-based baseline controller by approximately 8%, comparable to savings reported in the literature with MPC for similar DCEPs. We describe the design choices in the RL controller, including basis functions, reward function shaping, and learning algorithm parameters. Compared to existing work on RL for DCEPs, the proposed controller is designed for continuous state and actions spaces.
We discuss the application of a nonlinear model predictive control (MPC) and a moving horizon estimation (MHE) to achieve an optimal operation of \textit{E. coli} fed-batch cultivations with intermittent bolus feeding. 24 parallel experiments were considered in a high-throughput microbioreactor platform at a 10 mL scale. The robotic island in question can run up to 48 fed-batch processes in parallel with automated liquid handling and online and at-line analytics. The implementation of the model-based monitoring and control framework reveals that there are mainly three challenges that need to be addressed; First, the inputs are given in an instantaneous pulsed form by bolus injections, second, online and at-line measurement frequencies are severely imbalanced, and third, optimization for the distinctive multiple reactors can be either parallelized or integrated. We address these challenges by incorporating the concept of impulsive control systems, formulating multi-rate MHE with identifiability analysis, and suggesting criteria for deciding the reactor configuration. In this study, we present the key elements and background theory of the implementation with \textit{in silico} simulations for bacterial fed-batch cultivation.
In this paper, we present a data-driven model predictive control (MPC) scheme that is capable of stabilizing unknown linear time-invariant systems under the influence of process disturbances. To this end, Willems' lemma is used to predict the future behavior of the system. This allows the entire scheme to be set up using only a priori measured data and knowledge of an upper bound on the system order. First, we develop a state-feedback MPC scheme, based on input-state data, which guarantees closed-loop practical exponential stability and recursive feasibility as well as closed-loop constraint satisfaction. The scheme is extended by a suitable constraint tightening, which can also be constructed using only data. In order to control a priori unstable systems, the presented scheme contains a pre-stabilizing controller and an associated input constraint tightening. We first present the proposed data-driven MPC scheme for the case of full state measurements, and also provide extensions for obtaining similar closed-loop guarantees in case of output feedback. The presented scheme is applied to a numerical example.
Stochastic model predictive control has been a successful and robust control framework for many robotics tasks where the system dynamics model is slightly inaccurate or in the presence of environment disturbances. Despite the successes, it is still unclear how to best adjust control parameters to the current task in the presence of model parameter uncertainty and heteroscedastic noise. In this paper, we propose an adaptive MPC variant that automatically estimates control and model parameters by leveraging ideas from Bayesian optimisation (BO) and the classical expected improvement acquisition function. We leverage recent results showing that BO can be reformulated via density ratio estimation, which can be efficiently approximated by simply learning a classifier. This is then integrated into a model predictive path integral control framework yielding robust controllers for a variety of challenging robotics tasks. We demonstrate the approach on classical control problems under model uncertainty and robotics manipulation tasks.
Motivated by the recently proposed dual control for exploration and exploitation (DCEE) concept, this paper presents a Multi-Step DCEE (MS-DCEE) framework with guaranteed convergence for autonomous search of a source of airborne dispersion. Different from the existing stochastic model predictive control (SMPC) algorithm and informative path planning (IPP) approaches, the proposed MS-DCEE approach uses the current and future input to not only drive the agent towards the estimated source location (exploitation) but also reduce its estimation uncertainty (exploration) by actively learning the operational environment. Unknown source target position, together with unknown environment, impose significant challenges in establishing the recursive feasibility and the convergence of the proposed algorithm. To address them, with the help of the property of Bayesian estimation, we develop a two-step approach where the unbiasedness of the mean estimation is assumed first and then the randomness of the mean estimate under each collected information sequence is accounted. Based on that, we develop a MS-DCEE scheme with suitable terminal ingredients where recursive feasibility and convergence are guaranteed. Two simulation scenarios are conducted, which show that the proposed MS-DCEE algorithm outperforms the SMPC, the IPP and the single-step DCEE approaches in terms of searching successful rates and efficiency.
Data-driven model predictive control has two key advantages over model-free methods: a potential for improved sample efficiency through model learning, and better performance as computational budget for planning increases. However, it is both costly to plan over long horizons and challenging to obtain an accurate model of the environment. In this work, we combine the strengths of model-free and model-based methods. We use a learned task-oriented latent dynamics model for local trajectory optimization over a short horizon, and use a learned terminal value function to estimate long-term return, both of which are learned jointly by temporal difference learning. Our method, TD-MPC, achieves superior sample efficiency and asymptotic performance over prior work on both state and image-based continuous control tasks from DMControl and Meta-World. Code and video results are available at https://nicklashansen.github.io/td-mpc.
This study presents a Bayesian learning perspective towards model predictive control algorithms. High-level frameworks have been developed separately in the earlier studies on Bayesian learning and sampling-based model predictive control. On one hand, the Bayesian learning rule provides a general framework capable of generating various machine learning algorithms as special instances. On the other hand, the dynamic mirror descent model predictive control framework is capable of diversifying sample-rollout-based control algorithms. However, connections between the two frameworks have still not been fully appreciated in the context of stochastic optimal control. This study combines the Bayesian learning rule point of view into the model predictive control setting by taking inspirations from the view of understanding model predictive controller as an online learner. The selection of posterior class and natural gradient approximation for the variational formulation governs diversification of model predictive control algorithms in the Bayesian learning approach to model predictive control. This alternative viewpoint complements the dynamic mirror descent framework through streamlining the explanation of design choices.
A variety of control tasks such as inverse kinematics (IK), trajectory optimization (TO), and model predictive control (MPC) are commonly formulated as energy minimization problems. Numerical solutions to such problems are well-established. However, these are often too slow to be used directly in real-time applications. The alternative is to learn solution manifolds for control problems in an offline stage. Although this distillation process can be trivially formulated as a behavioral cloning (BC) problem in an imitation learning setting, our experiments highlight a number of significant shortcomings arising due to incompatible local minima, interpolation artifacts, and insufficient coverage of the state space. In this paper, we propose an alternative to BC that is efficient and numerically robust. We formulate the learning of solution manifolds as a minimization of the energy terms of a control objective integrated over the space of problems of interest. We minimize this energy integral with a novel method that combines Monte Carlo-inspired adaptive sampling strategies with the derivatives used to solve individual instances of the control task. We evaluate the performance of our formulation on a series of robotic control problems of increasing complexity, and we highlight its benefits through comparisons against traditional methods such as behavioral cloning and Dataset aggregation (Dagger).
LEO satellite constellation coverage has served as the base platform for various space applications. However, the rapidly evolving security environment such as orbit debris and adversarial space threats are greatly endangering the security of satellite constellation and integrity of the satellite constellation coverage. As on-orbit repairs are challenging, a distributed and autonomous protection mechanism is necessary to ensure the adaptation and self-healing of the satellite constellation coverage from different attacks. To this end, we establish an integrative and distributed framework to enable resilient satellite constellation coverage planning and control in a single orbit. Each satellite can make decisions individually to recover from adversarial and non-adversarial attacks and keep providing coverage service. We first provide models and methodologies to measure the coverage performance. Then, we formulate the joint resilient coverage planning-control problem as a two-stage problem. A coverage game is proposed to find the equilibrium constellation deployment for resilient coverage planning and an agent-based algorithm is developed to compute the equilibrium. The multi-waypoint Model Predictive Control (MPC) methodology is adopted to achieve autonomous self-healing control. Finally, we use a typical LEO satellite constellation as a case study to corroborate the results.
We propose a stochastic MPC scheme using an optimization over the initial state for the predicted trajectory. Considering linear discrete-time systems under unbounded additive stochastic disturbances subject to chance constraints, we use constraint tightening based on probabilistic reachable sets to design the MPC. The scheme avoids the infeasibility issues arising from unbounded disturbances by including the initial state as a decision variable. We show that the stabilizing control scheme can guarantee constraint satisfaction in closed loop, assuming unimodal disturbances. In addition to illustrating these guarantees, the numerical example indicates further advantages of optimizing over the initial state for the transient behavior.
The problem of synthesizing stochastic explicit model predictive control policies is known to be quickly intractable even for systems of modest complexity when using classical control-theoretic methods. To address this challenge, we present a scalable alternative called stochastic parametric differentiable predictive control (SP-DPC) for unsupervised learning of neural control policies governing stochastic linear systems subject to nonlinear chance constraints. SP-DPC is formulated as a deterministic approximation to the stochastic parametric constrained optimal control problem. This formulation allows us to directly compute the policy gradients via automatic differentiation of the problem's value function, evaluated over sampled parameters and uncertainties. In particular, the computed expectation of the SP-DPC problem's value function is backpropagated through the closed-loop system rollouts parametrized by a known nominal system dynamics model and neural control policy which allows for direct model-based policy optimization. We provide theoretical probabilistic guarantees for policies learned via the SP-DPC method on closed-loop stability and chance constraints satisfaction. Furthermore, we demonstrate the computational efficiency and scalability of the proposed policy optimization algorithm in three numerical examples, including systems with a large number of states or subject to nonlinear constraints.
We present a stochastic model predictive control (SMPC) framework for linear systems subject to possibly unbounded disturbances. State of the art SMPC approaches with closed-loop chance constraint satisfaction recursively initialize the nominal state based on the previously predicted nominal state or possibly the measured state under some case distinction. We improve these initialization strategies by allowing for a continuous optimization over the nominal initial state in an interpolation of these two extremes. The resulting SMPC scheme can be implemented as one standard quadratic program and is more flexible compared to state-of-the-art initialization strategies. As the main technical contribution, we show that the proposed SMPC framework also ensures closed-loop satisfaction of chance constraints and suitable performance bounds.
Distributed model predictive control methods for uncertain systems often suffer from considerable conservatism and can tolerate only small uncertainties due to the use of robust formulations that are amenable to distributed design and optimization methods. In this work, we propose a distributed stochastic model predictive control (DSMPC) scheme for dynamically coupled linear discrete-time systems subject to unbounded additive disturbances that are potentially correlated in time. An indirect feedback formulation ensures recursive feasibility of the DSMPC problem, and a data-driven, distributed and optimization-free constraint tightening approach allows for exact satisfaction of chance constraints during closed-loop control, addressing typical sources of conservatism. The computational complexity of the proposed controller is similar to nominal distributed MPC. The approach is demonstrated in simulation for the temperature control of a large-scale data center subject to randomly varying computational loads.