Model predictive control (MPC) has become a hot cake technology for various applications due to its ability to handle multi-input multi-output systems with physical constraints. The optimization solvers require considerable time, limiting their embedded implementation for real-time control. To overcome the bottleneck of traditional quadratic programming (QP) solvers, this paper proposes a robust penalty method (RPM) to solve an optimization problem in a linear MPC. The main idea of RPM is to solve an unconstrained QP problem using Broyden Fletcher Goldfarb Shannon (BFGS) algorithm. The beauty of this method is that it can find optimal solutions even if initial conditions are in an infeasible region, which makes it robust. Moreover, the RPM is computationally inexpensive as compared to the traditional QP solvers. The proposed RPM is implemented on resource-limited embedded hardware (STM32 microcontroller), and its performance is validated with a case study of a citation aircraft control problem. We show the hardware-in-the-loop co-simulation results of the proposed RPM and compared them with the active set method (ASM) and interior point method (IPM) QP solvers. The performance of MPC with the aforementioned solvers is compared by considering the optimality, time complexity, and ease of hardware implementation. Presented results show that the proposed RPM gives the same optimality as ASM and IPM, and outperforms them in terms of speed.
In this paper, we consider a stochastic Model Predictive Control able to account for effects of additive stochastic disturbance with unbounded support, and requiring no restrictive assumption on either independence nor Gaussianity. We revisit the rather classical approach based on penalty functions, with the aim of designing a control scheme that meets some given probabilistic specifications. The main difference with previous approaches is that we do not recur to the notion of probabilistic recursive feasibility, and hence we do not consider separately the unfeasible case. In particular, two p…
Real-world reinforcement learning (RL) problems often demand that agents behave safely by obeying a set of designed constraints. We address the challenge of safe RL by coupling a safety guide based on model predictive control (MPC) with a modified policy gradient framework in a linear setting with continuous actions. The guide enforces safe operation of the system by embedding safety requirements as chance constraints in the MPC formulation. The policy gradient training step then includes a safety penalty which trains the base policy to behave safely. We show theoretically that this penalty al…
It is very well-known that the implementation of Model Predictive Controller (MPC) on embedded platforms is challenging due to the computational complexities associated while solving an optimization problem. Although, there are many efficient embedded implementations existing by now, but for faster, more dynamic and non-linear control applications, there is no cost-effective and memory efficient embedded solutions. In this paper, we show the implementation of embedded explicit MPC for a motor speed control application on a lowcost 8 bit PIC 18 series microcontroller which costs only $5. The of…
This article presents a sparse, low-memory footprint optimization algorithm for the implementation of the model predictive control (MPC) for tracking formulation in embedded systems. This MPC formulation has several advantages over standard MPC formulations, such as an increased domain of attraction and guaranteed recursive feasibility even in the event of a sudden reference change. However, this comes at the expense of the addition of a small amount of decision variables to the MPC's optimization problem that complicates the structure of its matrices. We propose a sparse optimization algorith…
Optimal scheduling of deferrable electrical loads can reshape the aggregated load profile to achieve higher operational efficiency and reliability. This paper studies deferrable load scheduling under demand charge that imposes a penalty on the peak consumption over a billing period. Such a terminal cost poses challenges in real-time dispatch when demand forecasts are inaccurate. A block model-predictive control approach is proposed by breaking demand charge into a sequence of stage costs. The problem of charging electric vehicles is used to illustrate the efficacy of the proposed approach. Num…
We introduce the notion of importance sampling under embedded barrier state control, titled Safety Controlled Model Predictive Path Integral Control (SC-MPPI). For robotic systems operating in an environment with multiple constraints, hard constraints are often encoded utilizing penalty functions when performing optimization. Alternative schemes utilizing optimization-based techniques, such as Control Barrier Functions, can be used as a safety filter to ensure the system does not violate the given hard constraints. In contrast, this work leverages the principle of a safety filter but applies i…
We propose a fully distributed control system architecture, amenable to in-vehicle implementation, that aims to safely coordinate connected and automated vehicles (CAVs) at road intersections. For control purposes, we build upon a fully distributed model predictive control approach, in which the agents solve a nonconvex optimal control problem (OCP) locally and synchronously, and exchange their optimized trajectories via vehicle-to-vehicle (V2V) communication. To accommodate a fast solution of the nonconvex OCPs, we apply the penalty convex-concave procedure which solves a convexified version …
Multi-Agent Temporal Logic Planning via Penalty Functions and Block-Coordinate Optimization
University West（韦斯特大学）；Max Planck Institute for Software Systems（马克斯·普朗克软件系统研究所）；ETH Zürich（苏黎世联邦理工学院）；Queen’s University Belfast（贝尔法斯特女王大学）；KTH Royal Institute of Technology（瑞典皇家理工学院）
The cost of the power distribution infrastructures is driven by the peak power encountered in the system. Therefore, the distribution network operators consider billing consumers behind a common transformer in the function of their peak demand and leave it to the consumers to manage their collective costs. This management problem is, however, not trivial. In this paper, we consider a multi-agent residential smart grid system, where each agent has local renewable energy production and energy storage, and all agents are connected to a local transformer. The objective is to develop an optimal pol…
We present a differentiable predictive control (DPC) methodology for learning constrained control laws for unknown nonlinear systems. DPC poses an approximate solution to multiparametric programming problems emerging from explicit nonlinear model predictive control (MPC). Contrary to approximate MPC, DPC does not require supervision by an expert controller. Instead, a system dynamics model is learned from the observed system's dynamics, and the neural control law is optimized offline by leveraging the differentiable closed-loop system model. The combination of a differentiable closed-loop syst…
The modular open-source framework GRAMPC-D for model predictive control of distributed systems is presented in this paper. The modular concept allows to solve optimal control problems (OCP) in a centralized and distributed fashion using the same problem description. It is tailored to computational efficiency with the focus on embedded hardware. The distributed solution is based on the Alternating Direction Method of Multipliers (ADMM) and uses the concept of neighbor approximation to enhance convergence speed. The presented framework can be accessed through Cpp and Python and also supports plu…
T2S-MPC: Time-Embedded Online Adaptive Model Predictive Control for Time-Varying Dynamics
JHU Department of Applied Mathematics and Statistics, Johns Hopkins University, MD, USA（约翰霍普金斯大学应用数学与统计学系）；CMU Department of Electrical and Computer Engineering, Carnegie Mellon University, PA, USA（卡内基梅隆大学电气与计算机工程系）
In this paper we present a dual active-set solver for quadratic programming which has properties suitable for use in embedded model predictive control applications. In particular, the solver is efficient, can easily be warm-started, and is simple to code. Moreover, the exact worst-case computational complexity of the solver can be determined offline and, by using outer proximal-point iterations, ill-conditioned problems can be handled in a robust manner.
This article presents the real-time implementation of the model predictive control for tracking formulation to control a two-wheeled inverted pendulum robot. This formulation offers several advantages over standard MPC formulations at the expense of the addition of a small number of decision variables, which complicates the inner structure of the matrices of the optimization problem. We implement a sparse solver, based on an extension of the alternating direction method of multipliers, in the system's embedded hardware. The results indicate that the solver is suitable for controlling a real sy…
This paper analyzes the stability-related properties of Long Short-Term Memory (LSTM) networks and investigates their use as the model of the plant in the design of Model Predictive Controllers (MPC). First, sufficient conditions guaranteeing the Input-to-State stability (ISS) and Incremental Input-to-State stability (dISS) of LSTM are derived. These properties are then exploited to design an observer with guaranteed convergence of the state estimate to the true one. Such observer is then embedded in a MPC scheme solving the tracking problem. The resulting closed-loop scheme is proved to be as…
For controller design for systems on manifolds embedded in Euclidean space, it is convenient to utilize a theory that requires a single global coordinate system on the ambient Euclidean space rather than multiple local charts on the manifold or coordinate-free tools from differential geometry. In this article, we apply such a theory to design model predictive tracking controllers for systems whose dynamics evolve on manifolds and illustrate its efficacy with the fully actuated rigid body attitude control system.
This paper presents a systematic method for the selection of the Model Predictive Control (MPC) stage cost. We match the MPC feedback law to a proportional-integral (PI) controller, which we efficiently tune by high-performance Monte Carlo (MC) simulation. The PI tuning offers a wide range of tuning possibilities that is then inherited by the MPC design. The MC simulation tuning of the PI controller is based on the minimization of two different objectives; 1) the 2-norm tracking error, and 2) a bi-objective consisting of the 2-norm tracking error and a 2-norm input rate of movement penalty. We…
This work explores the nature of augmented importance sampling in safety-constrained model predictive control problems. When operating in a constrained environment, sampling based model predictive control and motion planning typically utilizes penalty functions or expensive optimization based control barrier algorithms to maintain feasibility of forward sampling. In contrast the presented algorithm utilizes discrete embedded barrier states in augmented importance sampling to apply feedback with respect to a nominal state when sampling. We will demonstrate that this approach of safety of discre…
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, re…
We present differentiable predictive control (DPC) as a deep learning-based alternative to the explicit model predictive control (MPC) for unknown nonlinear systems. In the DPC framework, a neural state-space model is learned from time-series measurements of the system dynamics. The neural control policy is then optimized via stochastic gradient descent approach by differentiating the MPC loss function through the closed-loop system dynamics model. The proposed DPC method learns model-based control policies with state and input constraints, while supporting time-varying references and constrai…
Robotic Systems Lab, ETH Zurich（苏黎世联邦理工学院机器人系统实验室）
The Sequential Linear Quadratic (SLQ) algorithm is a continuous-time variant of the well-known Differential Dynamic Programming (DDP) technique with a Gauss-Newton Hessian approximation. This family of methods has gained popularity in the robotics community due to its efficiency in solving complex trajectory optimization problems. However, one major drawback of DDP-based formulations is their inability to properly incorporate path constraints. In this paper, we address this issue by devising a constrained SLQ algorithm that handles a mixture of constraints with a previously implemented project…
Real-time Neural-MPC: Deep Learning Model Predictive Control for Quadrotors and Agile Robotic Platforms
Munich Institute of Robotics and Machine Intelligence (MIRMI)（慕尼黑机器人与机器学习研究所（MIRMI））；Technical University of Munich（慕尼黑工业大学）；University of Zurich（苏黎世大学）；Stanford University（斯坦福大学）；NVIDIA Research（英伟达研究院）
Aggregators have emerged as crucial tools for the coordination of distributed, controllable loads. To be used effectively, an aggregator must be able to communicate the available flexibility of the loads they control, as known as the aggregate flexibility to a system operator. However, most of existing aggregate flexibility measures often are slow-timescale estimations and much less attention has been paid to real-time coordination between an aggregator and an operator. In this paper, we consider solving an online optimization in a closed-loop system and present a design of real-time aggregate…
School of Electrical and Computer Engineering（电气与计算机工程学院）；King Fahd University of Petroleum & Minerals（法赫德国王石油与矿产大学）；School of Computer Science（计算机科学学院）；George W. Woodruff School of Mechanical Engineering（乔治·W·伍德拉夫机械工程学院）；Daniel Guggenheim School of Aerospace Engineering（丹尼尔·古根海姆航空航天工程学院）
Certified safe control is a growing challenge in robotics, especially when performance and safety objectives must be concurrently achieved. In this work, we extend the barrier state (BaS) concept, recently proposed for safe stabilization of continuous time systems, to safety embedded trajectory optimization for discrete time systems using discrete barrier states (DBaS). The constructed DBaS is embedded into the discrete model of the safety-critical system integrating safety objectives into the system's dynamics and performance objectives. Thereby, the control policy is directly supplied by saf…
In this paper, we provide a stability and performance analysis of model predictive control (MPC) schemes based on finite-tail costs. We study the MPC formulation originally proposed by Magni et al. (2001) wherein the standard terminal penalty is replaced by a finite-horizon cost of some stabilizing control law. In order to analyse the closed loop, we leverage the more recent technical machinery developed for MPC without terminal ingredients. For a specified set of initial conditions, we obtain sufficient conditions for stability and a performance bound in dependence of the prediction horizon a…
We propose a new risk-constrained formulation of the classical Linear Quadratic (LQ) stochastic control problem for general partially-observed systems. Our framework is motivated by the fact that the risk-neutral LQ controllers, although optimal in expectation, might be ineffective under relatively infrequent, yet statistically significant extreme events. To effectively trade between average and extreme event performance, we introduce a new risk constraint, which explicitly restricts the total expected predictive variance of the state penalty by a user-prescribed level. We show that, under cer…
Koopman-Based Robust Model Predictive Control for Nonlinear Systems with Stochastic Intermittent Measurements
Harbin Institute of Technology（哈尔滨工业大学）；School of Astronautics, Harbin Institute of Technology（哈尔滨工业大学航天学院）；Nanyang Technological University（南洋理工大学）；Nanyang Environment and Water Research Institute (NEWRI)（南洋环境与水研究）
Code Generation and Conic Constraints for Model-Predictive Control on Microcontrollers with Conic-TinyMPC
School of Engineering and Applied Science, Columbia University（哥伦比亚大学工程与应用科学学院）；Carnegie Mellon University（卡内基梅隆大学）；Massachusetts Institute of Technology（麻省理工学院）；Barnard College, Columbia University and Dartmouth College（哥伦比亚大学巴纳德学院和达特茅斯学院）
