Non-Linear Model Predictive Control with Adaptive Time-Mesh Refinement

In this paper, we present a novel solution for real-time, Non-Linear Model Predictive Control (NMPC) exploiting a time-mesh refinement strategy. The proposed controller formulates the Optimal Control Problem (OCP) in terms of flat outputs over an adaptive lattice. In common approximated OCP solutions, the number of discretization points composing the lattice represents a critical upper bound for real-time applications. The proposed NMPC-based technique refines the initially uniform time horizon by adding time steps with a sampling criterion that aims to reduce the discretization error. This enables a higher accuracy in the initial part of the receding horizon, which is more relevant to NMPC, while keeping bounded the number of discretization points. By combining this feature with an efficient Least Square formulation, our solver is also extremely time-efficient, generating trajectories of multiple seconds within only a few milliseconds. The performance of the proposed approach has been validated in a high fidelity simulation environment, by using an UAV platform. We also released our implementation as open source C++ code.Comment: In: 2018 IEEE International Conference on Simulation, Modeling, and Programming for Autonomous Robots (SIMPAR 2018

A Simple and Efficient Algorithm for Nonlinear Model Predictive Control

We present PANOC, a new algorithm for solving optimal control problems arising in nonlinear model predictive control (NMPC). A usual approach to this type of problems is sequential quadratic programming (SQP), which requires the solution of a quadratic program at every iteration and, consequently, inner iterative procedures. As a result, when the problem is ill-conditioned or the prediction horizon is large, each outer iteration becomes computationally very expensive. We propose a line-search algorithm that combines forward-backward iterations (FB) and Newton-type steps over the recently introduced forward-backward envelope (FBE), a continuous, real-valued, exact merit function for the original problem. The curvature information of Newton-type methods enables asymptotic superlinear rates under mild assumptions at the limit point, and the proposed algorithm is based on very simple operations: access to first-order information of the cost and dynamics and low-cost direct linear algebra. No inner iterative procedure nor Hessian evaluation is required, making our approach computationally simpler than SQP methods. The low-memory requirements and simple implementation make our method particularly suited for embedded NMPC applications

Dominant speed factors of active set methods for fast MPC

The paper presents a review of active set (AS) algorithms that have been deployed for implementation of fast model predictive control (MPC). The main purpose of the survey is to identify the dominant features of the algorithms that contribute to fast execution of online MPC and to study their influence on the speed. The simulation study is conducted on two benchmark examples where the algorithms are analyzed in the number of iterations and in the workload per iteration. The obtained results suggest directions for potential improvement in the speed of existing AS algorithms

Predictive Maneuver Planning and Control of an Autonomous Vehicle in Multi-Vehicle Traffic with Observation Uncertainty

Autonomous vehicle technology is a promising development for improving the safety, efficiency and environmental impact of on-road transportation systems. However, the task of guiding an autonomous vehicle by rapidly and systematically accommodating the plethora of changing constraints, e.g. of avoiding multiple stationary and moving obstacles, obeying traffic rules, signals and so on as well as the uncertain state observation due to sensor imperfections, remains a major challenge. This dissertation attempts to address this challenge via designing a robust and efficient predictive motion planning framework that can generate the appropriate vehicle maneuvers (selecting and tracking specific lanes, and related speed references) as well as the constituent motion trajectories while considering the differential vehicle kinematics of the controlled vehicle and other constraints of operating in public traffic. The main framework combines a finite state machine (FSM)-based maneuver decision module with a model predictive control (MPC)-based trajectory planner. Based on the prediction of the traffic environment, reference speeds are assigned to each lane in accordance with the detection of objects during measurement update. The lane selection decisions themselves are then incorporated within the MPC optimization. The on-line maneuver/motion planning effort for autonomous vehicles in public traffic is a non-convex problem due to the multiple collision avoidance constraints with overlapping areas, lane boundaries, and nonlinear vehicle-road dynamics constraints. This dissertation proposes and derives some remedies for these challenges within the planning framework to improve the feasibility and optimality of the solution. Specifically, it introduces vehicle grouping notions and derives conservative and smooth algebraic models to describe the overlapped space of several individual infeasible spaces and help prevent the optimization from falling into undesired local minima. Furthermore, in certain situations, a forced objective selection strategy is needed and adopted to help the optimization jump out of local minima. Furthermore, the dissertation considers stochastic uncertainties prevalent in dynamic and complex traffic and incorporate them with in the predictive planning and control framework. To this end, Bayesian filters are implemented to estimate the uncertainties in object motions and then propagate them into the prediction horizon. Then, a pair-wise probabilistic collision condition is defined for objects with non-negligible geometrical shape/sizes and computationally efficient and conservative forms are derived to efficiently and analytically approximate the involved multi-variate integrals. The probabilistic collision evaluation is then applied within a vehicle grouping algorithms to cluster the object vehicles with closeness in positions and speeds and eventually within the stochastic predictive maneuver planner framework to tighten the chanced-constraints given a deterministic confidence margin. It is argued that these steps make the planning problem tractable for real-time implementation on autonomously controlled vehicles

Nonlinear Model Predictive Control Reduction Strategies for Real-time Optimal Control

This thesis presents a variety of strategies to accelerate the turnaround times (TATs) of nonlinear and hybrid model predictive controllers (MPCs). These strategies are unified by the themes of symbolic computing, nonlinear model reduction and automotive control. The first contribution of this thesis is a new MPC problem formulation, called symbolic single shooting (symSS), that leverages the power of symbolic computing to generate an optimization problem of minimal dimension. This formulation is counter to the recent trend of introducing and exploiting sparsity of the MPC optimization problem for tailored solvers to exploit. We make use of this formulation widely in this thesis. The second contribution of this thesis is a novel application of proper orthogonal decomposition (POD) to MPC. In this strategy we construct a dimensionally-reduced optimization problem by restricting the problem Lagrangian to a subspace. This subspace is found by running simulations offline from which we extract the important solution features. Using this restricted Lagrangian we are able to reduce the problem dimension dramatically, thus simplifying the linear solve. This leads to TAT accelerations of more than two times with minimal controller degradation. The third contribution of this thesis is an informed move blocking strategy. This strategy exploits the features extracted in the restricted Lagrangian subspace to derive a sequence of increasingly blocked move blocking strategies. These move blocking strategies can then be used to reduce the dimension of the optimization problem in a sparse manner, leading to even greater acceleration of the controller TAT . The fourth contribution of this thesis is a new quasi-Newton method for MPC. This method utilizes ideas similar to singular perturbation-based model reduction to truncate the expression for the problem Hessian at the symbolic level. For nonlinear systems with a modest Lipschitz constant, we can identify the timestep as a `small' parameter about which we can do a perturbative expansion of the Lagrangian and its derivatives. Truncating to first order in the timestep, we are able to find a good approximation of the Hessian leading to TAT acceleration. The fifth contribution of this thesis is controller integration strategy based on nested MPCs. Using the symSS formulation we can construct an explicit model of a controlled plant that includes the full model as well as the MPC's action. This form of the controlled plant model allows us to generate exact derivatives so that fast solvers can be used for real time application. We focus here on the problem of planning and motion control integration for autonomous vehicles but this strategy can be extended for other problems that require accurate models of a controlled plant. The sixth contribution of this thesis is a strategy to handle integer controls in MPC based on a few reasonable assumptions: our predictions over the horizon are almost perfect and the future is inevitable. These assumptions enforce a degree of continuity, in the integer controls, between solutions over different timesteps that allow us to mitigate chatter and enforce a hard upper bound on solution complexity. This strategy constrains the integer solution of one timestep to be related to that of the previous timestep. Our results show that this strategy provides acceptable control performance while achieving TATs that are orders of magnitude smaller than those for conventional MINLP-based methods, thereby opening the door to new real-time applications of hybrid MPC

Commande prédictive nonlinéaire d'un système de freinage hybride électro-hydraulique régénératif

Le contexte de cette étude porte sur l’implémentation d’une stratégie de commande collaborative entre le système de freinage électro-hydraulique d’un véhicule récréatif trois-roues à propulsion électrique et son système de freinage régénératif. Grâce à l’établissement d’une méthode originale d’allocation des efforts de freinage aux roues avant / arrière, la stratégie développée permet de spécifier la consigne d’effort de freinage « idéale » à la roue arrière en fonction du freinage hydraulique à l’avant. Une commande prédictive nonlinéaire a été développée pour répartir cette consigne entre le frein électro-hydraulique et le frein régénératif en temps réel. Cette stratégie de contrôle consiste à trouver la commande optimale à appliquer sur un horizon de temps glissant à durée finie. La commande est la solution d’un problème d’optimisation sous contraintes dures, dont le processus a été implémenté en-ligne avec la prise en charge directe d’un modèle de prédiction nonlinéaire des dynamiques rapides du système de freinage hybride. L’identification paramétrique et la validation de ce modèle ont été réalisées expérimentalement sous diverses conditions. L’analyse des résultats de la commande prédictive, tirés de simulations ainsi que d’expérimentations sous forme d’essais routiers montre des résultats conformes aux comportements attendus. En effet, la stratégie permet de maximiser le couple de freinage régénératif lors d’un freinage nécessitant l’utilisation collaborative des deux systèmes de frein tout en respectant la consigne de freinage. De plus, on observe des propriétés de robustesse à l’incertitude paramétrique. Finalement, les résultats expérimentaux démontrent qu’avec la croissance de la puissance de calcul embarquée, la commande prédictive nonlinéaire devient envisageable pour de nombreuses applications temps-réel à dynamique rapide

Algorithms and Applications for Nonlinear Model Predictive Control with Long Prediction Horizon

Fast implementations of NMPC are important when addressing real-time control of systems exhibiting features like fast dynamics, large dimension, and long prediction horizon, as in such situations the computational burden of the NMPC may limit the achievable control bandwidth. For that purpose, this thesis addresses both algorithms and applications. First, fast NMPC algorithms for controlling continuous-time dynamic systems using a long prediction horizon have been developed. A bridge between linear and nonlinear MPC is built using partial linearizations or sensitivity update. In order to update the sensitivities only when necessary, a Curvature-like measure of nonlinearity (CMoN) for dynamic systems has been introduced and applied to existing NMPC algorithms. Based on CMoN, intuitive and advanced updating logic have been developed for different numerical and control performance. Thus, the CMoN, together with the updating logic, formulates a partial sensitivity updating scheme for fast NMPC, named CMoN-RTI. Simulation examples are used to demonstrate the effectiveness and efficiency of CMoN-RTI. In addition, a rigorous analysis on the optimality and local convergence of CMoN-RTI is given and illustrated using numerical examples. Partial condensing algorithms have been developed when using the proposed partial sensitivity update scheme. The computational complexity has been reduced since part of the condensing information are exploited from previous sampling instants. A sensitivity updating logic together with partial condensing is proposed with a complexity linear in prediction length, leading to a speed up by a factor of ten. Partial matrix factorization algorithms are also proposed to exploit partial sensitivity update. By applying splitting methods to multi-stage problems, only part of the resulting KKT system need to be updated, which is computationally dominant in on-line optimization. Significant improvement has been proved by giving floating point operations (flops). Second, efficient implementations of NMPC have been achieved by developing a Matlab based package named MATMPC. MATMPC has two working modes: the one completely relies on Matlab and the other employs the MATLAB C language API. The advantages of MATMPC are that algorithms are easy to develop and debug thanks to Matlab, and libraries and toolboxes from Matlab can be directly used. When working in the second mode, the computational efficiency of MATMPC is comparable with those software using optimized code generation. Real-time implementations are achieved for a nine degree of freedom dynamic driving simulator and for multi-sensory motion cueing with active seat