Engineered cyberphysical systems are growing increasingly large and complex. These systems require scalable controllers that robustly satisfy state and input constraints in the presence of additive noise -- such controllers should also be accompanied by theoretical guarantees on feasibility and stability. In our companion paper, we introduced Distributed and Localized Model Predictive Control (DLMPC) for large-scale linear systems; DLMPC is a scalable closed-loop MPC scheme in which subsystems need only exchange local information in order to synthesize and implement local controllers. In this paper, we provide recursive feasibility and asymptotic stability guarantees for DLMPC. We leverage the System Level Synthesis framework to express the maximal positive robust invariant set for the closed-loop system and its corresponding Lyapunov function, both in terms of the closed-loop system responses. We use the invariant set as the terminal set for DLMPC, and show that this guarantees feasibility with minimal conservatism. We use the Lyapunov function as the terminal cost, and show that this guarantees stability. We provide fully distributed and localized algorithms to compute the terminal set offline, and also provide necessary additions to the online DLMPC algorithm to accommodate coupled terminal constraint and cost. In all algorithms, only local information exchanges are necessary, and computational complexity is independent of the global system size -- we demonstrate this analytically and experimentally. This is the first distributed MPC approach that provides minimally conservative yet fully distributed guarantees for recursive feasibility and asymptotic stability, for both nominal and robust settings.
An explicit Model Predictive Control algorithm for large-scale structured linear systems is presented. We base our results on Distributed and Localized Model Predictive Control (DLMPC), a closed-loop model predictive control scheme based on the System Level Synthesis (SLS) framework wherein only local state and model information needs to be exchanged between subsystems for the computation and implementation of control actions. We provide an explicit solution for each of the subproblems resulting from the distributed MPC scheme. We show that given the separability of the problem, the explicit s…
We propose a tube-based guaranteed cost model predictive controller considering a homothetic formulation for constrained linear systems subject to multiplicative structured norm-bounded uncertainties. It provides an upper bound to the general min-max model predictive control. The invariance property of the proposed tube holds for any arbitrary scaling. It yields a second-order cone programming problem which is less computationally expensive than standard semi-definite programming problems. We also present a numerical example with a comparative study among the proposed approach, an open-loop gu…
We present the Distributed and Localized Model Predictive Control (DLMPC) algorithm for large-scale structured linear systems, wherein only local state and model information needs to be exchanged between subsystems for the computation and implementation of control actions. We use the System Level Synthesis (SLS) framework to reformulate the MPC problem as an optimization problem over closed loop system responses, and show that this allows us to naturally impose localized communication constraints between sub-controllers, such that only local state and system model information needs to be excha…
Motivated by large-scale but computationally constrained settings, e.g., the Internet of Things, we present a novel data-driven distributed control algorithm that is synthesized directly from trajectory data. Our method, data-driven Distributed and Localized Model Predictive Control (D$^3$LMPC), builds upon the data-driven System Level Synthesis (SLS) framework, which allows one to parameterize \emph{closed-loop} system responses directly from collected open-loop trajectories. The resulting model-predictive controller can be implemented with distributed computation and only local information s…
We provide a comprehensive review and practical implementation of a recently developed model predictive control (MPC) framework for controlling unknown systems using only measured data and no explicit model knowledge. Our approach relies on an implicit system parametrization from behavioral systems theory based on one measured input-output trajectory. The presented MPC schemes guarantee closed-loop stability for unknown linear time-invariant (LTI) systems, even if the data are affected by noise. Further, we extend this MPC framework to control unknown nonlinear systems by continuously updating…
The development of control techniques to maintain vehicle stability under possible loss-of-control scenarios is essential to the safe deployment of autonomous ground vehicles in public scenarios. In this paper, we propose a tube-based guaranteed cost model predictive controller for autonomous vehicles able to avoid front and rear tire saturation and to track a provided reference trajectory up to the limits of handling of the vehicle. Such an approach ensures the vehicle will remain within its safe operational envelope; therefore, guaranteeing both stability and performance of the vehicle, incl…
We present a data-driven model predictive control scheme for chance-constrained Markovian switching systems with unknown switching probabilities. Using samples of the underlying Markov chain, ambiguity sets of transition probabilities are estimated which include the true conditional probability distributions with high probability. These sets are updated online and used to formulate a time-varying, risk-averse optimal control problem. We prove recursive feasibility of the resulting MPC scheme and show that the original chance constraints remain satisfied at every time step. Furthermore, we show…
We propose Kernel Predictive Control (KPC), a learning-based predictive control strategy that enjoys deterministic guarantees of safety. Noise-corrupted samples of the unknown system dynamics are used to learn several models through the formalism of non-parametric kernel regression. By treating each prediction step individually, we dispense with the need of propagating sets through highly non-linear maps, a procedure that often involves multiple conservative approximation steps. Finite-sample error bounds are then used to enforce state-feasibility by employing an efficient robust formulation. …
In this paper we present a Learning Model Predictive Control (LMPC) strategy for linear and nonlinear time optimal control problems. Our work builds on existing LMPC methodologies and it guarantees finite time convergence properties for the closed-loop system. We show how to construct a time varying safe set and terminal cost function using closed-loop data. The resulting LMPC policy is time varying and it guarantees recursive constraint satisfaction and non-decreasing performance. Computational efficiency is obtained by convexifing the safe set and terminal cost function. We demonstrate that,…
In this paper we propose a novel decision making architecture for Robust Model Predictive Path Integral control (RMPPI) and investigate its performance guarantees and applicability to off-road navigation. Key building blocks of the proposed architecture are an augmented state space representation of the system consisting of nominal and actual dynamics, a placeholder for different types of tracking controllers, a safety logic for nominal state propagation, and an importance sampling scheme that takes into account the capabilities of the underlying tracking control. Using these ingredients, we d…
Pacific Northwest National Laboratory（太平洋西北国家实验室）；Battelle Memorial Institute（巴特尔纪念研究所）
We develop a novel form of differentiable predictive control (DPC) with safety and robustness guarantees based on control barrier functions. DPC is an unsupervised learning-based method for obtaining approximate solutions to explicit model predictive control (MPC) problems. In DPC, the predictive control policy parametrized by a neural network is optimized offline via direct policy gradients obtained by automatic differentiation of the MPC problem. The proposed approach exploits a new form of sampled-data barrier function to enforce offline and online safety requirements in DPC settings while …
We consider the effect of using approximate system predictions in event-triggered control schemes. Such approximations may result from using numerical transcription methods for solving continuous-time optimal control problems. Mesh refinement can guarantee upper bounds on the error in the differential equations which model the system dynamics. With the accuracy guarantees of a mesh refinement scheme, we show that the proposed event-triggering scheme -- which compares the measured system with approximate state predictions -- can be used with a guaranteed strictly positive inter-update time. We …
Distributed model predictive control (DMPC) is often used to tackle path planning for unmanned aerial vehicle (UAV) swarms. However, it requires considerable computations on-board the UAV, leading to increased weight and power consumption. In this work, we propose to offload path planning computations to multiple ground-based computation units. As simultaneously communicating and recomputing all trajectories is not feasible for a large swarm with tight timing requirements, we develop a novel event-triggered DMPC that selects a subset of most relevant UAV trajectories to be replanned. The resul…
To effectively control large-scale distributed systems online, model predictive control (MPC) has to swiftly solve the underlying high-dimensional optimization. There are multiple techniques applied to accelerate the solving process in the literature, mainly attributed to software-based algorithmic advancements and hardware-assisted computation enhancements. However, those methods focus on arithmetic accelerations and overlook the benefits of the underlying system's structure. In particular, the existing decoupled software-hardware algorithm design that naively parallelizes the arithmetic oper…
Automated vehicles require efficient and safe planning to maneuver in uncertain environments. Largely this uncertainty is caused by other traffic participants, e.g., surrounding vehicles. Future motion of surrounding vehicles is often difficult to predict. Whereas robust control approaches achieve safe, yet conservative motion planning for automated vehicles, Stochastic Model Predictive Control (SMPC) provides efficient planning in the presence of uncertainty. Probabilistic constraints are applied to ensure that the maximal risk remains below a predefined level. However, safety cannot be ensur…
We introduce a general framework for robust data-enabled predictive control (DeePC) for linear time-invariant (LTI) systems. The proposed framework enables us to obtain model-free optimal control for LTI systems based on noisy input/output data. More specifically, robust DeePC solves a min-max optimization problem to compute the optimal control sequence that is resilient to all possible realizations of the uncertainties in the input/output data within a prescribed uncertainty set. We present computationally tractable reformulations of the min-max problem with various uncertainty sets. Furtherm…
We present differentiable predictive control (DPC), a method for learning constrained neural control policies for linear systems with probabilistic performance guarantees. We employ automatic differentiation to obtain direct policy gradients by backpropagating the model predictive control (MPC) loss function and constraints penalties through a differentiable closed-loop system dynamics model. We demonstrate that the proposed method can learn parametric constrained control policies to stabilize systems with unstable dynamics, track time-varying references, and satisfy nonlinear state and input …
Otto-von-Guericke-University（奥托·冯·格里克大学）；Universidad de Sevilla（塞维利亚大学）
Model predictive control allows to provide high performance and safety guarantees in the form of constraint satisfaction. These properties, however, can be satisfied only if the underlying model, used for prediction, of the controlled process is sufficiently accurate. One way to address this challenge is by data-driven and machine learning approaches, such as Gaussian processes, that allow to refine the model online during operation. We present a combination of an output feedback model predictive control scheme and a Gaussian process-based prediction model that is capable of efficient online l…
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 illu…
ReLU Networks for Model Predictive Control: Network Complexity and Performance Guarantees
Department of Automation, and Beijing National Research Center for Info. Sci. & Tech. (BNRist), Tsinghua University, Beijing 100084, China（自动化系，北京信息科学与技术国家研究中心（BNRist），清华大学，北京）
We propose a robust data-driven model predictive control (MPC) scheme to control linear time-invariant (LTI) systems. The scheme uses an implicit model description based on behavioral systems theory and past measured trajectories. In particular, it does not require any prior identification step, but only an initially measured input-output trajectory as well as an upper bound on the order of the unknown system. First, we prove exponential stability of a nominal data-driven MPC scheme with terminal equality constraints in the case of no measurement noise. For bounded additive output measurement …
Network-Realised Model Predictive Control Part II: Distributed Constraint Management
Department of Automatic Control；Systems Engineering, ``Politehnica'' University of Bucharest, Bucharest, Romania -1mm；Laboratoire des Signaux et Syst\`emes, CentraleSup\'elec, Paris-Saclay University, Gif-sur-Yvette, France -9mm
In this paper, a novel tube-based economic Model Predictive Control (MPC) scheme for uncertain systems that uses neither terminal costs nor terminal constraints is investigated. We show that the results from the undisturbed case can be extended to systems with bounded disturbances by using similar turnpike arguments and a properly modified stage cost. We prove robust guarantees on the closed-loop performance, convergence, and stability under suitable dissipativity and controllability conditions and discuss them in a numerical example.
