Online MPC Research Articles

Hybrid Machine Learning Forecasting for Online MPC of Work Place Electric Vehicle Charging

Hybrid Machine Learning Forecasting for Online MPC of Work Place Electric Vehicle Charging

Evaluation of optimisation techniques on the performance of an on-line MPC controller in an occupancy grid for autonomous mobile robots

Computational capabilities on embedded platforms, especially for autonomous mobile robotics, enable them to be used for more challenging tasks, e.g., to apply advanced control methods like an on-line MPC controller on the robot itself. In the last decades, many improvements in direct methods to solve the optimal control problem were proposed, which speed up the optimisation process. We apply the multiple shooting approach to solve the Optimal Control Problem in a parallel manner. We apply the on-line MPC scheme with the multiple shooting approach in simulations, assess them in a hardware-in-the-loop scenario and conduct experiments on a Raspberry Pi 4-based differential-drive-platform to assess the realtime capability to control the robot while the optimisation is carried out locally.

Model Predictive Control for Lane Merging Automation with Recursive Feasibility Guarantees

In order to make the complex driving task of merging safer, in this paper we consider the automated merging of an autonomous vehicle into a mixed-traffic flow scenario (i.e., traffic including autonomous and manually driven vehicles). In particular, we propose a novel MPC-based algorithm to perform a merging procedure from a double lane into a single lane and continue with (adaptive) cruise control ((A)CC) functionality after the merge. The proposed MPC balances fast progress along the path with comfort, while obeying safety and maximum allowed velocity bounds. Recursive feasibility, leading to safety and proper behavior, is guaranteed by the design of a proper terminal set, extending existing ones in the literature. The on-line MPC problem is translated into a mixed integer quadratic program (MIQP) that can be solved for global optimality. Through numerical simulations we demonstrate the behavior and effectiveness of the proposed MPC merging scheme.

Practical Model Predictive Control for a Class of Nonlinear Systems Using Linear Parameter-Varying Representations

In this paper, a practical <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">model predictive control</i> (MPC) for tracking desired reference trajectories is demonstrated for controlling a class of nonlinear systems subject to constraints, which comprises diverse mechanical applications. Owing to the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">linear parameter-varying</i> (LPV) formulation of the associated nonlinear dynamics, the online MPC optimization problem is solvable as a single <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">quadratic programming</i> (QP) problem of complexity similar to that of LTI systems. For offset-free tracking, based on the notion of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">admissible reference</i> , the controller ensures convergence to any admissible reference while its deviation from the desired reference is penalized in the stage cost of the optimization problem. This mechanism provides a safety feature under the physical limitations of the system. To guarantee stability and recursive feasibility, a terminal cost as a tracking error penalty term and a terminal constraint associated with both the terminal state and the admissible reference are included. We use tube-based concept to deal with the uncertainty of the scheduling parameter over the prediction horizon. Therefore, the online optimization problem is solved for only the nominal system corresponding to the current value of the scheduling parameter and subject to tightened constraint sets. The proposed approach has been implemented successfully in real-time onto a robotic manipulator, the experimental results illustrates its efficiency and practicality.

Robust model predictive control for a nanofluid based solar thermal power plant

Among the several technologies for solar energy recovery, parabolic solar collectors have emerged as one of the most promising due to their performance, which can be enhanced if nanofluids are employed as heat transfer fluids instead of the traditional alternatives. The inherent time-dependent behavior of solar radiation profiles forces the solar thermal plants to be operated aided with controllers able to reject these strong disturbances. While traditional controllers can be employed for this aim, more advanced techniques such as Model Predictive Control are suggested since this optimal-control based method can be tuned to minimize operating costs, among some other features. The main objective of this work is to implement an MPC controller to a nanofluid-based solar thermal power plant in order to evaluate its performance to reject disturbances on the solar radiation profile in an efficient manner. An off-line nonlinear programming optimization was deployed so we could compare the response of the on-line MPC implementation on a strict enough basis. Furthermore, the performance of MPC controllers is affected by how well does the modeling of the system is able to stick to reality, thus, it is important to test if the controller is robust enough to deal with uncertainty that might be introduced as modeling errors. Results indicate that MPC controllers are suitable for their implementation on these kinds of power plants since they steer the system to achieve desired conditions by smoothly manipulating the decision variable, even in the scenarios where a substantial cascade-effect modeling error was imposed in the parameters of the nanofluid.

Identification of a dynamic system model for a building and heating system including heat pump and thermal energy storage

Controllers employing optimal control strategies will path the way to enable flexible operations in future power grids. As buildings will increasingly act as prosumers in future power grids, optimal control of buildings’ energy consumption will play a major role in providing flexible operations. Optimal controllers such as model predictive controller are able to manage buildings’ operations and to optimise their energy consumption. For online optimisation, model predictive controller requires a model of the energy system. The more accurate the system model represents the system dynamics, the more accurate the model predictive controller predicts the future states of the energy system while optimising its energy consumption. In this article, we present a system model that can be used in online MPC, including dynamic programming as optimisation strategy. The system model is validated using a building and heating system, including heat pump and thermal energy storage.The following bullet points summarise the main requirements for the configuration of the system model:•The system model performs fast with low computational effort in less than 1 s;•The system model can be implemented in online MPC;•The system model accurately represents the dynamic behaviour.

FPGA Implementation Framework for Explicit Hybrid Model Predictive Control

Hybrid systems can be controlled by a Hybrid Model Predictive Control (HMPC) with an accurate prediction model and proper formulation. In the real-time evolution of HMPC, an online optimization (mostly, Mixed Integer Quadratic Programming (MIQP)) problem needs to be solved at each sample time. Online implementation of MIQP solver on resource-limited embedded platforms poses challenges, which limits its use for the slow dynamic systems. The online burden of MIQP solver can be moved offline by multi-parametric programming. In this paper, we present an implementation of explicit HMPC on Field Programmable Gate Array (FPGA) for the control of Mixed Logical Dynamical (MLD) systems. We show the step-by-step procedure of FPGA implementation of HMPC considering the model of the inverted pendulum. In the implementation, we used MATLAB-based Hybrid Toolbox to construct hybrid MPC and subsequently generate a low-level C/C++ code of the HMPC. Further, the generated C/C++ code is optimized for memory, speed, and resource utilization by the customized approach of applying pipelining and directives using Xilinx Vivado. The customized explicit HMPC is then implemented on a Xilinx ZYNQ-7000 SoC ZC706 FPGA. The implemented approach is compared with the linear (online and explicit) MPC with a linearized model of the system. The detailed analysis of controller computational complexity in terms of memory, resource utilization, clock, and power consumption is presented. The performance of implemented HMPC is verified through Hardware-in-the-Loop (HIL) co-simulations.

Multiparametric Nonlinear MPC: A region free approach

Determination of active constraints forms an essential part of the multiparametric MPC approach for linear systems. An analysis of KKT conditions to identify active constraints provides piecewise affine control laws and their corresponding critical regions (CRs). However, an extension of multiparametric MPC for nonlinear systems requires overcoming significant challenges: predictions are nonlinear and so are constraints, in which case the MPC problems takes the form of a nonlinear program (NLP). Application of KKT conditions show that, in general, the MPC control law for nonlinear systems is piecewise, implicit, nonlinear function of the state. Moreover, the CRs have nonlinear boundaries. In this work, we propose an offline combinatorial approach to identify all active sets of constraints for the nonlinear MPC problem a priori. The offline approach uses implicit enumeration of the constraints based on feasibility of KKT conditions and a primal criteria. Thus, the offline step provides all the admissible CRs as well as the corresponding nonlinear system of KKT equations corresponding to each CR. The online MPC implementation uses a region-free approach, wherein the CR corresponding to the current state as well as the control action is determined by solving the nonlinear system of KKT equations online. The method is demonstrated using a numerical example from literature.

Predictive Control for Networked Interval Type-2 T–S Fuzzy System via an Event-Triggered Dynamic Output Feedback Scheme

In this paper, the problem of event-triggered dynamic output feedback model predictive control (OFMPC) for nonlinear networked control systems (NCSs) with packet loss and bounded disturbance is studied. Interval type-2 (IT2) Takagi–Sugeno fuzzy model is exploited to represent the nonlinear plant with parameter uncertainties, which can be captured by the lower and upper membership functions. Whether or not the measured output should be released into unreliable network links is determined by the error between the current measured output and the latest event-triggered output. The Bernoulli random binary distribution is used to describe the process of packet loss in NCSs. This paper proposes the following: 1) the synthesis approach, including the design of the parameter-dependent dynamic output feedback controller by solving an online MPC optimization problem, which minimizes the upper bound of an infinite time horizon quadratic objective function respecting input and state constraints; 2) the guarantee of recursive feasibility and quadratical stability of the closed-loop system by applying the quadratic boundedness technique. Moreover, an algorithm of tightening the ellipsoidal bounds of state error is added to improve the control performance. The simulation and comparison studies are performed to demonstrate the usefulness and availability of the presented new techniques.

Low Complexity Distributed Model Predictive Control by Using Contractive Sets

In this paper, an approach to low complexity distributed MPC of linear interconnected systems with coupled dynamics subject to both state and input constraints is proposed. The suggested approach is based on the idea of introducing a contractive set constraint in the centralized MPC problem formulation, which would guarantee the closed-loop system stability when using a small prediction horizon. Then, a dual accelerated gradient method is applied to obtain distributedly a suboptimal solution of the resulting Quadratic Programming problem. The suggested approach would be appropriate for embedded distributed MPC since it will reduce the complexity of the on-line MPC computations, simplify the software implementation, and reduce the requirements for available memory.

Alternative LMI-based Robust MPC Design Approaches

The paper presents several alternative approaches of on-line robust MPC design based on LMI formulation. The proposed approaches are based on existing approaches. Suitable properties of these approaches were adopted to reduce the conservativeness of the quadratic stability condition, to reduce the control inputs constraints evaluation, and to minimize a computational effort. Therefore these alternative robust MPC methods may be considered as a tool for overcoming some of robust MPC design obstacles.

Interpolation-based Off-line Robust MPC for Uncertain Polytopic Discrete-time Systems

In this paper, interpolation-based off-line robust MPC for uncertain polytopic discrete-time systems is presented. Instead of solving an on-line optimization problem at each sampling time to find a state feedback gain, a sequence of state feedback gains is pre-computed off-line in order to reduce the on-line computational time. At each sampling time, the real-time state feedback gain is calculated by linear interpolation between the pre-computed state feedback gains. Three interpolation techniques are proposed. In the first technique, the smallest ellipsoids containing the measured state are approximated and the corresponding real-time state feedback gain is calculated. In the second technique, the pre-computed state feedback gains are interpolated in order to get the largest possible real-time state feedback gain while robust stability is still guaranteed. In the last technique, the real-time state feedback gain is calculated by minimizing the violation of the constraints of the adjacent inner ellipsoids so the real-time state feedback gain calculated has to regulate the state from the current ellipsoids to the adjacent inner ellipsoids as fast as possible. As compared to on-line robust MPC, the proposed techniques can significantly reduce on-line computational time while the same level of control performance is still ensured.

N-step off-line MPC design of nonhomogeneous Markov jump systems: A suboptimal case

This paper mainly concerns N-step off-line suboptimal predictive controller design for discrete nonhomogeneous Markov jump systems, in which the Markov chains are time-varying transition probabilities matrix modeled as a polytope. The design procedure is divided into N-step, more precisely, the first is to design the Nth step when the changes of Euclidean form of mode-dependent feedback law between the Nth and the (N+1)th asymptotically stable mode-dependent ellipsoids are less than the given accuracy. Then the Nth asymptotically stable mode-dependent invariant ellipsoid is defined. In the previous (N−1) steps, an off-line mode-dependent predictive controller is designed to drive the state to this small area including the origin. Compared with on-line MPC algorithm, the computation time is dramatically reduced while the dynamic performance of controller is comparable. One numerical example is presented to illustrate the validity of the developed results.

Multi Reference Model Predictive EGR-Valve Control for Diesel Engines

Accurate control of the engine air system is essential to comply with emission legislation. Typically the real emission targets, like NOx, cannot be measured directly at a production car and thus different intermediate candidates, like gas mass flows or oxygen concentrations, are used. In general the candidate choice is a key factor for emissions, because each of them can only be determined with certain accuracy and consequently a feedback control based on a specific quantity is only capable of reaching a point within the accuracy range. Moreover, a slight deviation of one candidate may, due to the nonlinear relations between emissions and candidates, lead to high NOx deviations. In a previous work four candidates for EGR valve control were analyzed and a scheduling scheme according to their NOx tracking performance under uncertainties was proposed. Within this work the approach is extended by the application of a multi reference online MPC control strategy which takes also dynamical properties into account. To this end a scheduling between the candidates based on the static and dynamic analysis in the previous work is applied, where the idea is to adapt the inherited objective function during runtime. The proposed dynamic reference scheduling strategy was implemented on a simulation model of a passenger car Diesel engine and satisfying results were obtained.

Explicit weight selection procedure for optimal control problems with weighted objectives

AbstractMany practical optimal control problems (e.g., (Nonlinear) Model Predictive Control ((N)MPC)) involve multiple and conflicting objectives. Most often, this multi-objective nature is tackled by combining all individual objectives into a global weighted sum. However, selecting appropriate weights is a non-trivial task, especially when no price information is available and/or the different objectives are incommensurable. This paper introduces a strategy that exploits recent scalarisation based multi-objective optimisation methods to generate the set of Pareto optimal solutions from which the decision maker can pick a solution according to his/her preferences. This selected solution can explicitly be linked to a set of weights in a weighted sum approach via analytical relations. These weights can then, e.g., be used for the repeated optimisations in online (N)MPC applications. The proposed selection procedure based on the explicit relations is illustrated for the offline optimal control of a nonlinear jacketed tubular reactor with three conflicting objectives: conversion, net heat transfer and reactor length.

Performance Measures in Model Predictive Control with Non-linear Prediction Horizon Time-discretization

Model predictive sampled-data control of constrained, linear, time-invariant, continuous-time plants is considered. The time-discretization of the prediction horizon may be non-linear, in order to reduce the computational complexity of online MPC methods by lowering the number of optimization variables for a given prediction horizon length. The main contribution of this paper is to propose two closed-loop performance measures in order to evaluate the salient performance properties of non-linearly time-discretized prediction horizons. A numerical motivating example comparing two prediction horizon time-discretizations with an order of magnitude difference in the number of optimization variables is discussed, and subsequently the results of a sensitivity analysis of the two proposed performance measures with respect to the prediction horizon time-discretization are presented. The use of non-linearly time-discretized prediction horizons is also shown to be relevant for complexity reduction in offline MPC strategies.