Nominal Model Predictive Control Research Articles

On the Inherent Distributional Robustness of Stochastic and Nominal Model Predictive Control

We define a notion of distributional robustness, via the Wasserstein metric, for closed-loop systems subject to errors in the disturbance distribution used to construct the controller. We then establish sufficient conditions for stochastic model predictive control (SMPC) to satisfy this definition of distributional robustness and establish a similar notion of distributional robustness for economic applications of SMPC. These results address incorrectly or unmodeled disturbances, demonstrate the efficacy of scenario optimization as a means to approximate and solve the SMPC problem, and unify the descriptions of robustness for stochastic and nominal model predictive control. This definition of distributional robustness for closed-loop systems is general and can be applied to other stochastic optimal control algorithms and, potentially, the developing field of distributionally robust control.

Fourth-Order Suboptimality of Nominal Model Predictive Control in the Presence of Uncertainty

Fourth-Order Suboptimality of Nominal Model Predictive Control in the Presence of Uncertainty

Closed-loop optimisation of neural networks for the design of feedback policies under uncertainty

Solving model predictive control (MPC) problems online can be computationally intractable, especially when considering uncertainty and nonlinear systems. One approach to avoid this is to train a neural network on a data-set of solutions of MPC problems (potentially nonlinear) offline, and to evaluate the trained control policy online. However, due to the separation of these optimisation problems the neural network controller must be carefully verified to ensure adequate closed loop performance. In this paper we propose a method to train a neural network in closed loop for control systems, in continuous or discrete time, while allowing for flexible consideration of parametric uncertainty. This method does not require off-line solutions of the NMPC problem, and instead directly optimises the desired closed loop performance. We prove that our method can approximate the optimal closed-loop control policy to arbitrary tolerance and in numerical examples demonstrate its performance compared to explicit MPC, imitation learning and nominal MPC.

A practically implementable reinforcement learning‐based process controller design

AbstractThe present article enables reinforcement learning (RL)‐based controllers for process control applications. Existing instances of RL‐based solutions have significant challenges for online implementation since the training process of an RL agent (controller) presently requires practically impossible number of online interactions between the agent and the environment (process). To address this challenge, we propose an implementable model‐free RL method developed by leveraging industrially implemented model predictive control (MPC) calculations (often designed using a simple linear model identified via step tests). In the first step, MPC calculations are used to pretrain an RL agent that can mimic the MPC performance. Specifically, the MPC calculations are used to pretrain the actor, and the objective function is used to pretrain the critic(s). The pretrained RL agent is then employed within a model‐free RL framework to control the process in a way that initially imitates MPC behavior (thus not compromising process performance and safety), but also continuously learns and improve its performance over the nominal linear MPC. The effectiveness of the proposed approach is illustrated through simulations on a chemical reactor example.

The Stochastic Robustness of Nominal and Stochastic Model Predictive Control

In this work, we establish and compare the stochastic and deterministic robustness properties achieved by nominal model predictive control (MPC), stochastic MPC (SMPC), and a proposed constraint-tightened MPC (CMPC) formulation, which represents an idealized version of tube-based MPC. We consider three definitions of robustness for nonlinear systems and bounded disturbances: robust asymptotic stability (RAS), robust asymptotic stability in expectation (RASiE), and RASiE w.r.t. the stage cost <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell (\cdot)$</tex-math></inline-formula> used in these MPC formulations ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell$</tex-math></inline-formula> -RASiE). Via input-to-state stability (ISS) and stochastic ISS (SISS) Lyapunov functions, we establish that MPC, subject to sufficiently small disturbances, and CMPC ensure all three definitions of robustness without a stochastic objective function. While SMPC also ensures RASiE and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell$</tex-math></inline-formula> -RASiE, SMPC does not guarantee RAS for nonlinear systems. Through a few simple examples, we illustrate the implications of these results and demonstrate that, depending on the definition of robustness considered, SMPC is not necessarily more robust than nominal MPC even if the disturbance model is exact.

Robust MPC-based trajectory tracking of autonomous underwater vehicles with model uncertainty

A robust model predictive control (MPC) method with dual closed-loops is presented to handle trajectory tracking of autonomous underwater vehicle (AUV) with uncertain model parameters and random external perturbations. First, constraint conditions are set for the motion state and control input of the underwater vehicle based on its motion characteristics. The position controller takes the velocity increment as input, thus providing a smoothly varying desired velocity for the velocity controller. The velocity controller comprises nominal MPC and a nonlinear auxiliary control law to overcome the effect of random perturbations on AUV tracking control. Then, a finite-time extended state observer (FTESO) is designed to compensate for dynamic model uncertainty. Furthermore, Lyapunov stability theory is employed to analyse the stability of the controller and FTESO. Ultimately, through comparative simulation experiments, the proposed control framework's effectiveness and robustness are verified, proving it to be a feasible AUV trajectory tracking control method.

DGPR‐MPC: Learning‐based model predictive controller for autonomous vehicle path following

AbstractIn this paper, the authors investigate the issue of constructing and incorporating an accurate vehicle dynamic model for model predictive control (MPC) with an application to autonomous vehicle path following. To achieve the desired performance, MPC employs a precise dynamic model. However, the uncertainty of complex systems and their operating environments presents a challenge to the development of an adequately accurate vehicle dynamic model. This paper proposes a Deep Gaussian Process Regression (DGPR) method to improve model precision. Meanwhile, the learning model is incorporated into a novel MPC framework to enhance closed‐loop performance. High‐fidelity simulations using CarSim‐MATLAB demonstrate the validity of the proposed approach in terms of enhancing the path following performance and lateral stability under the condition of large curvature at medium to high speeds on roads with different friction coefficients when compared to the nominal MPC approach.

An Improved Model Predictive Control-Based Trajectory Planning Method for Automated Driving Vehicles Under Uncertainty Environments

For automated driving vehicles, trajectory planning is responsible for obtaining feasible trajectories with velocity profiles according to driving environments. From the perspective of trajectory planning, multiple uncertainties of environments and tracking deviations are two significant factors affecting driving safety. The former disturbs the judgment of trajectory planning on the environments, and the latter reduces the tracking accuracy of planned trajectories. To solve these problems, an improved model predictive control (MPC) trajectory planning method is proposed in this paper. Firstly, a Kalman filter fusion method is carried out to predict obstacle trajectory and their uncertainty, which combines model-based and data-based prediction methods. Based on the prediction results, a tube-based MPC trajectory planning method is applied to plan a reference trajectory with a small tracking deviation. The tube-based MPC is composed of two parts. One is the MPC with tightened constraints that is used to plan a feasible trajectory according to a nominal vehicle system and driving environment. The other is a state feedback control that is proposed to adjust the above planned trajectory to reduce the tracking deviations. To our knowledge, this paper proposes Kalman filter fusion and tube-based MPC planning method for the first time to consider the uncertainties of trajectory prediction and tracking control meanwhile in the planning. The planning method is verified by simulations and experiments in multiple scenes. Results show that the method is suitable for both static and dynamic scenes. Compared with applying the basic prediction method, the lateral deviation of the proposed method from the ideal trajectory is decreased by 46.5%. Compared with the nominal MPC method, the lateral tracking deviations of the proposed method are decreased by 77.42%.

A decentralized robust control approach for virtually coupled train sets

AbstractIn this paper, a novel approach to train control systems based on virtual coupling is presented. Virtual coupling is a concept that has evolved from platooning of vehicles and allows to reduce the distance and headway between trains without constructing new lines while ensuring safe operation. With this objective in mind, we propose a decentralized robust model predictive control (MPC) framework for a virtually coupled train set based on a min–max approach. Unlike the nominal MPC, robust MPC is designed to consider external undetermined disturbances and errors to improve robustness in real‐world applications. Therefore, in this study, we present the formulation of a robust MPC based on solving a finite‐horizon optimization problem with bounded uncertainties. The bounds consider resistive modeling errors, positioning errors, communication delays, and a possible adhesion loss of up to 10%. We then performed four simulations to compare the behavior of the robust MPC with the equivalent nominal MPC. In these simulations, we simulated a metro line, main line, and high‐speed line. The simulations also analyzed the behavior of the robust MPC under the considered perturbations and different communication delays. The results show that the robust MPC ensures safer operation than nominal MPC in subways, conventional lines, and high‐speed lines. Future research can focus on centralized MPC and artificial intelligence.

Efficient Uncertainty Mitigation in Automated Vehicles: Combining MPC with Model Reference Adaptive Control

Automated vehicles must be equipped with the ability to plan and execute trajectories in the presence of uncertainties for safe and effective navigation in both on-and of-road environments. Model predictive control (MPC) is a powerful and popular optimal control solution for both planning and control tasks, but even robust MPC solutions still suffer from decreased performance and the curse of dimensionality in the optimization problem when subject to model errors. We propose an adaptive control scheme that combines a fast, nominal MPC with our previously developed Model Reference Adaptive Control strategy in a cascaded architecture. The effectiveness of the resulting framework is demonstrated via robust and computationally efficient trajectory tracking under severe uncertainties. This study includes detailed tuning considerations and numerical evaluation with different fidelity vehicle models.

Heteroscedastic Bayesian Optimisation for Active Power Control of Wind Farms*

Active power control of wind farms remains an open challenge due to inherent noise in wind power that arises from uncertain wind speed measurements and plant/model mismatch. To leverage the heteroscedastic nature of the wind power noise, heteroscedastic Bayesian optimisation (BO) is used for active power control of wind farms. BO utilises closed-loop performance data to tune the parameters of a stochastic model predictive controller (SMPC) in a systematic and data-efficient manner. This, in turn, allows for enhancing the closed-loop performance of the controller intended to decrease the power tracking error. A case study with 9 turbines in a 3 × 3 wind farm shows that the heteroscedastic BO approach achieves a reduced closed-loop power tracking error in terms of root-mean-square by 8.89% compared to one that relies on nominal BO and a decrease by 64.99% compared to a nominal model predictive controller (MPC) whose performance is not tuned using closed-loop data and BO.

Constrained control for systems on matrix Lie groups with uncertainties

AbstractIn this paper, the constrained control of systems evolving on matrix Lie groups with uncertainties is considered. The proposed methodology is composed of a nominal Model Predictive Control (MPC), and a feedback controller. The previous work on the control of systems on manifolds is applied to design the nominal MPC, which generates the nominal trajectory. In the nominal MPC, the state and input constraints on the Lie group are transformed into the constraints on the Euclidean space. While to deal with uncertainties, the feedback control used to track the nominal trajectory is designed directly on the Lie group. The tracking error in the feedback control is proved to be bounded in invariant sets, which are further used to revise the constraints in nominal MPC. We prove that the input‐to‐state stability of the entire system under the proposed control methodology with respect to the disturbances can be achieved. The proposed methodology is applied to the constrained attitude control of rigid bodies with uncertainties. In the application example, the detailed mathematical proof and the comparative numerical simulation are presented to demonstrate the feasibility of the proposed methodology.

Robust Learning-Based Predictive Control for Discrete-Time Nonlinear Systems With Unknown Dynamics and State Constraints

Robust model predictive control (MPC) is a well-known control technique for model-based control with constraints and uncertainties. In classic robust tube-based MPC approaches, an open-loop control sequence is computed via periodically solving an online nominal MPC problem, which requires prior model information and frequent access to onboard computational resources. In this paper, we propose an efficient robust MPC solution based on receding horizon reinforcement learning, called r-LPC, for unknown nonlinear systems with state constraints and disturbances. The proposed r-LPC utilizes a Koopman operator-based prediction model obtained off-line from pre-collected input-output datasets. Unlike classic tube-based MPC, in each prediction time interval of r-LPC, we use an actor-critic structure to learn a near-optimal feedback control policy rather than a control sequence. The resulting closed-loop control policy can be learned off-line and deployed online or learned online in an asynchronous way. In the latter case, online learning can be activated whenever necessary; for instance, the safety constraint is violated with the deployed policy. The closed-loop recursive feasibility, robustness, and asymptotic stability are proven under function approximation errors of the actor-critic networks. Simulation and experimental results on two nonlinear systems with unknown dynamics and disturbances have demonstrated that our approach has better or comparable performance when compared with tube-based MPC and LQR, and outperforms a recently developed actor-critic learning approach.

Closed-loop robust steady-state target calculation for model predictive control

Industrial model predictive control (MPC) typically includes a steady-state target calculation (SSTC) layer that provides targets for the dynamic MPC controller to track. This paper develops the first closed-loop robust SSTC method, which explicitly addresses the closed-loop uncertainty, for stable linear systems with time-invariant parametric uncertainty. The problem is initially formulated as a stochastic bi-level optimization problem, where the lower-level optimization problem describes the nominal MPC law. The problem is then transformed into a deterministic mixed-integer second-order cone program for ellipsoidal uncertainty. Case study results show that the proposed method is more robust than the classical nominal SSTC in that it helps reduce cycling of the steady-state targets. In addition, the proposed method is less conservative than a typical open-loop robust SSTC method, in that it drives the system to the steady state more quickly and can push the steady state closer to the operating boundary when desired.

Scenario Model Predictive Control for Data-Based Energy Management in Plug-In Hybrid Electric Vehicles

One of the major limitations of optimization-based strategies for allocating the power flow in hybrid powertrains is that they rely on predictions of future power demand. These predictions are inherently uncertain as they are dependent on complex human behaviours that are challenging to model accurately. This paper proposes a data-based scenario model predictive control framework, where the inputs are determined at each control update by optimizing the power allocation over multiple previous examples of a route being driven. The proposed energy management optimization is convex, and results from scenario optimization are used to bound the confidence that the one-step-ahead optimization will be feasible with given probability. It is shown through numerical simulation that scenario model predictive control (MPC) obtains the same reduction in fuel consumption as nominal MPC with full preview of future driver behaviour, and that the scenario MPC optimization can be solved efficiently using a tailored optimization algorithm.

Robust data-driven predictive control using reachability analysis

We present a robust data-driven control scheme for an unknown linear system model with bounded process and measurement noise. Instead of depending on a system model in traditional predictive control, a controller utilizing data-driven reachable regions is proposed. The data-driven reachable regions are based on a matrix zonotope recursion and are computed based on only noisy input-output data of a trajectory of the system. We assume that measurement and process noise are contained in bounded sets. While we assume knowledge of these bounds, no knowledge about the statistical properties of the noise is assumed. In the noise-free case, we prove that the presented purely data-driven control scheme results in an equivalent closed-loop behavior to a nominal model predictive control scheme. In the case of measurement and process noise, our proposed scheme guarantees robust constraint satisfaction, which is essential in safety-critical applications. Numerical experiments show the effectiveness of the proposed data-driven controller in comparison to model-based control schemes.

Robust tube-based model predictive control with Koopman operators

Koopman operators are of infinite dimension and capture the characteristics of nonlinear dynamics in a lifted global linear manner. The finite data-driven approximation of Koopman operators results in a class of linear predictors, useful for formulating linear model predictive control (MPC) of nonlinear dynamical systems with reduced computational complexity. However, the robustness of the closed-loop Koopman MPC under modeling approximation errors and possible exogenous disturbances is still a crucial issue to be resolved. Aiming at the above problem, this paper presents a robust tube-based MPC solution with Koopman operators, i.e., r-KMPC, for nonlinear discrete-time dynamical systems with additive disturbances. The proposed controller is composed of a nominal MPC using a lifted Koopman model and an off-line nonlinear feedback policy. The proposed approach does not assume the convergence of the approximated Koopman operator, which allows using a Koopman model with a limited order for controller design. Fundamental properties, e.g., stabilizability, observability, of the Koopman model are derived under standard assumptions with which, the closed-loop robustness and nominal point-wise convergence are proven. Simulated examples are illustrated to verify the effectiveness of the proposed approach.

Robust Path Tracking and Obstacle Avoidance using Tube-based Model Predictive Control for Surface Vehicles

This paper describes a tube-based model predictive control (TMPC) approach to robust path tracking and obstacle avoidance for surface vehicles. The TMPC algorithm consists of a nominal model predictive control and a state feedback control, which ensure that the state and input constraints are always satisfied under uncertain environmental disturbances. For obstacle avoidance, a robust positively invariant (RPI) set that contains the vehicle's position must be effectively calculated. To this end, the vehicle's dynamics are decoupled into surge and sway-yaw subsystems based on the vehicle's error dynamics with regard to the desired path, and zonotopes are used for the RPI set calculation. A TMPC is then designed using the obtained RPI set for each subsystem. Simulation results are presented to verify the effectiveness of the proposed control algorithm.

An Adaptive Model Predictive Control System for Virtual Coupling in Metros

Virtual coupling (VC) is an emerging concept and hot research topic in railways, especially for metro systems. Several unit trains in VC drive with a desired minimum distance, and they, as a whole, are regarded as a single train. In this work, a distributed adaptive model predictive control (AMPC) system is proposed to coordinate the driving of each unit train in VC. To obtain the accurate parameters of train dynamics model in a time varying environment, an estimator of the train dynamics model is designed for each AMPC controller. A variable step descent algorithm along the negative gradient direction is adopted for each estimator, which steers the estimated values of the parameters to real ones. Simulations are conducted and the results are compared with both nominal model predictive control system and AMPC system with fixed steps in the literature. Our proposed AMPC system with variable step (AMPCVS) has better performances than other two systems. Results indicate that there is an improvement of the proposed AMPC system with variable steps system when compared with other two existed systems. A running process of VC in a whole inter-station is also simulated here. Experimental results show that the trains track the desired objective well.

Disturbance-Observer-Based Model Predictive Control of Underwater Vehicle Manipulator Systems

Abstract This paper presents a disturbance-observer-based Model Predictive Control (MPC) for Underwater-Vehicle Manipulators Systems (UVMSs). First, the nominal MPC is formulated considering the lumped uncertainties representation for unknown terms and external disturbances. Then, a second-order Sliding Mode Disturbance Observer (SMDO) is developed to estimate the lumped disturbances, which are used by the MPC to produce unbiased predictions. The MPC-SMDO is tested through numerical simulations, considering a 5-DoF planar UVMS composed of the classical Twin-Burger autonomous vehicle presented in Ishitsuka et al. (2005) endowed with a 2-link robotic manipulator. To obtain realistic results, sensor noises, the dynamics of thrusters, and the stochasticity of ocean current are considered in the simulations. The results show good performance for the MPC-SMDO in terms of robustness, constraints meeting, and tracking errors minimization compared to two other controllers, based on the Computed Torque Control (CTC) technique and on the Super-Twisting Algorithm (STA).