Tube-based Robust Model Predictive Control Research Articles

An anti-disturbance lane-changing trajectory tracking control method combining extended Kalman filter and robust tube-based model predictive control

This paper proposes a trajectory tracking control method combining extended Kalman filter (EKF) and robust tube-based model predictive control (RTMPC) methods to improve the anti-disturbance capability during lane-changing maneuver of automated vehicles. A time-based quintic polynomial function is introduced for the implementation of trajectory planning to obtain the desired reference trajectory. The planned trajectory is input to the nominal system-oriented model predictive controller (MPC) in RTMPC for optimization to obtain the optimal control of the nominal system. The EKF collects the state measurements of the current instant and the optimal state estimates of the previous instant, and performs filtering to obtain the optimal state estimates of the current instant. The optimal estimate of the current state and the current state of the nominal system are input into the auxiliary control law of RTMPC to obtain the control of the actual system. Comparative numerical simulation experiments are conducted to analyze robustness and sensitivity of the proposed method. The results show that the control method combining EKF and RTMPC can optimize the trajectory tracking performance of the vehicle system, especially in the lateral displacement and the yaw-rate control. When the amplitude of measurement noise reaches the maximum, the optimization effect of lateral control is most significant in experiments. And the optimization effect in the control of lateral displacement and yaw angle continues to enhance with the increase of measurement disturbance. Therefore, this study can provide a reference for the anti-interference lane change trajectory tracking strategy of automated vehicles in the future.

Tube-Based Robust Model Predictive Control for Tracking Control of Autonomous Articulated Vehicles

Tube-Based Robust Model Predictive Control for Tracking Control of Autonomous Articulated Vehicles

Safe Motion Planning and Control Framework for Automated Vehicles with Zonotopic TRMPC

Model mismatches can cause multi-dimensional uncertainties for the receding horizon control strategies of automated vehicles (AVs). The uncertainties may lead to potentially hazardous behaviors when the AV tracks ideal trajectories that are individually optimized by the AV's planning layer. To address this issue, this study proposes a safe motion planning and control (SMPAC) framework for AVs. For the control layer, a dynamic model including multi-dimensional uncertainties is established. A zonotopic tube-based robust model predictive control scheme is proposed to constrain the uncertain system in a bounded minimum robust positive invariant set. A flexible tube with varying cross-sections is constructed to reduce the controller conservatism. For the planning layer, a concept of safety sets, representing the geometric boundaries of the ego vehicle and obstacles under uncertainties, is proposed. The safety sets provide the basis for the subsequent evaluation and ranking of the generated trajectories. An efficient collision avoidance algorithm decides the desired trajectory through the intersection detection of the safety sets between the ego vehicle and obstacles. A numerical simulation and hardware-in-the-loop experiment validate the effectiveness and real-time performance of the SMPAC. The result of two driving scenarios indicates that the SMPAC can guarantee the safety of automated driving under multi-dimensional uncertainties.

Tube-based robust model predictive control for fault tolerance

Tube-based robust model predictive control for fault tolerance

Uncertainties and Output Feedback in Rollout Event-Triggered Control

The fact that event-triggered control (ETC) often exhibits an improved performance-communication tradeoff over time-triggered control renders it especially useful for Networked Control Systems (NCSs). However, it has proven difficult to characterize the traffic produced by ETC a priori. Rollout ETC addresses this issue by using a triggering and control law that is <i>implicitly</i> defined by the solution to an optimal control problem (OCP), instead of an explicit one as in classical ETC. This allows to directly incorporate predefined constraints on the transmission traffic as well as on states and inputs. In this article, we examine the practically relevant case when output instead of state measurements are available, and measurements as well as the LTI plant are subject to uncertainties. To address these challenges, we adapt methods from robust tube-based model predictive control and propose three different strategies to implement an error feedback in an NCSs setup, the applicability of which depends on the capabilities of the actuator. We establish recursive feasibility, robust constraint satisfaction and convergence. Finally, we illustrate our results in a numerical example.

All-in-One Control Framework for Distributed Drive Electric Buses Path Tracking Subject to Uncertain Crosswind and Varied Passenger Mass

This paper proposes an all-in-one control framework for distributed drive electric buses (DDEBs) that can effectively attenuate the effects of uncertain crosswind and varied passenger mass in path tracking, while minimizing the electricity consumption of battery. To track the desired path and ensure driving stability, a robust tube-based model predictive control method is proposed considering lateral, yaw and roll motions. Simultaneously, for the longitudinal motion, a robust sliding mode control method is designed to achieve accurate tracking of longitudinal velocity. Meanwhile, the uncertain crosswind and varied passenger mass faced by the DDEBs are decoupled into external disturbances and their effects on the system are eliminated in both controllers. Then, to cope with energy optimization problem in path tracking, a torque distribution controller is constructed by formulating the vehicle stability margin, tracking dynamic requirements and energy efficiency as constrained weighted least squares problem. Finally, a Trucksim-Simulink co-simulation is performed, where the simulation results demonstrated that the proposed control framework is robust and effective, and can achieve better energy saving effect compared to the original control methods.

On the design of efficient optimal tube‐based robust model predictive control: Quasi‐H∞ approach

AbstractThis paper proposes a tube‐based robust model predictive control (TMPC) scheme with an optimal tube for disturbance‐affected linear systems. In the literature on TMPC, there is no proper methodology to handle the considerable effects of the tube size on the closed‐loop system performance. There is usually a trade‐off between the disturbance rejection level and the amount of control effort available for the MPC problem. In some applications, it is nearly impossible to find a feasible TMPC to have a sufficient amount of states and inputs feasible sets for the MPC optimization problem. It would be a vital contribution to the TMPC designs if an algorithm is demonstrated which can investigate the suitability of TMPC for a specific system. This paper provides a solution for the mentioned challenges by introducing the concept of Quasi‐H∞ criterion and proposing a constrained optimization problem. The optimization problem is then reformulated and simplified to present an efficient methodology for the TMPC designers. The proposed TMPC scheme could benefit from a larger terminal region and result in a larger region of attraction. The achievements in TMPC designs are shown by simulations and comparisons with the previously used techniquesover numerical case studies.

Energy saving and indoor temperature control for an office building using tube-based robust model predictive control

Actively controlling a building’s heating, ventilation, and air conditioning (HVAC) system can reduce costs and improve indoor comfort. Model predictive control (MPC) is an effective control algorithm that can facilitate the active control of complex systems such as the HVAC system. However, the uncertainty of the prediction model engenders many challenges in practical application. To address these issues, we propose a tube-based MPC strategy. First, a reduced-order thermal capacitance and thermal resistance model is established for the target system. Subsequently, a tube-based MPC scheme is designed to effectively handle uncertainties in real systems. The prediction uncertainty space is re-assumed in the tube, combined with the actual prediction error, to more closely correspond to the actual situation. The proposed model is tested and validated using the BOPTEST open-source testing framework. The results show that the proposed tube-based MPC can reduce the operating cost by at least 24%, compared with the traditional open-loop and closed-loop MPC, and can better control the indoor temperature when considering multiple uncertain predictions.

Robust Tube-Based Model Predictive Control for Wave Energy Converters

This paper proposes an efficient robust tube-based model predictive control (RTMPC) strategy for energy-maximization control of wave energy converters (WECs) subject to constraints due to safety considerations. Compared with the existing MPC strategies developed for the WEC control problem, the RTMPC method provides an effective approach to explicitly handle plant-model mismatches with guaranteed constraint satisfaction, contributing to improved energy capture efficiency. The fundamental idea is to integrate disturbance invariant sets into the MPC scheme for energy-maximization control to form a tube-based predictive controller, which enhances the robustness of MPC for a WEC without increasing online computational complexity. The resulting RTMPC controller can bound the WEC plant trajectories in a tube centered around a nominal WEC model trajectory, and uncertainties from un-modeled WEC dynamics and unmeasured disturbances can be mitigated by an error feedback portion. Numerical simulations demonstrate the effectiveness of the proposed control strategy.

Tube-Based Robust MPC for Two-Timescale Systems Using Reduced-Order Models

A tube-based robust Model Predictive Control (MPC) formulation is presented for systems with slow and fast timescale dynamics. The controller uses a reduced-order model that approximates the slow timescale dynamics and is implemented with a relatively large time step size. By analyzing the error between the full- and reduced-order models, the MPC optimization problem is proven to be recursively feasible and to produce closed-loop trajectories that satisfy state and input constraints. By relating bounds on all model errors to bounds on the change of the input in time, input change bounds are included as decision variables in the MPC problem to achieve a time-varying balance between fast transients with large model error and steady-state operation with zero model error. Zonotopes are used to make the approach practical and a numerical example demonstrates the benefits of optimizing time-varying input change bounds as part of the MPC formulation.

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.

Hierarchical MPC for coupled subsystems using adjustable tubes

A hierarchical Model Predictive Control (MPC) formulation is presented for coupled discrete-time linear systems with state and input constraints. Compared to a centralized approach, a two-level hierarchical controller, with one controller in the upper-level and one controller per subsystem in the lower-level, can significantly reduce the computational cost associated with MPC. Hierarchical coordination is achieved using adjustable tubes, which are optimized by the upper-level controller and bound permissible lower-level controller deviations from the system trajectories determined by the upper-level controller. The size of these adjustable tubes determines the degree of uncertainty between subsystems and directly affects the required constraint tightening under a tube-based robust MPC framework. Sets are represented as zonotopes to enable the ability to optimize the size of these adjustable tubes and perform the necessary constraint tightening online as part of the MPC optimization problems. State and input constraint satisfaction is proven for the two-level hierarchical controller with an arbitrary number of controllers at the lower-level and a numerical example demonstrates the key features and performance of the approach.

A Hierarchical Architecture for Optimal Unit Commitment and Control of an Ensemble of Steam Generators

A hierarchical architecture for the optimal management of an ensemble of steam generators is presented. The subsystems are coordinated by a multilayer scheme for jointly sustaining a common load. The high level optimizes the load allocation and the generator schedule, considering activation dynamics by a hybrid model. At the medium level, a robust tube-based model predictive control (MPC) tracks a time-varying demand using a centralized—but aggregate—model, whose order does not scale with the number of subsystems. A nonlinear optimization, at medium level, addresses MPC infeasibility due to abrupt changes of ensemble configuration. Low-level decentralized controllers stabilize the generators. This control scheme enables the dynamical modification of the ensemble configuration and plug and play operations. Simulations demonstrate the approach potentialities.

Varying Zonotopic tube RMPC with switching logic for lateral path tracking of autonomous vehicle

Model mismatch caused by strong nonlinearity and other factors will severely impact the lateral path tracking control in Autonomous Vehicles (AVs) under extreme conditions. Previous studies have focused on guaranteeing robust stability under possible uncertainty realizations through Tube-based Robust Model Predictive Control (TRMPC). However, three deficiencies in TRMPC applications are revealed: unknown disturbance set, simple rigid tube, and excessive conservatism. In this paper, a novel scheme named Varying Zonotopic TRMPC with Switching Logic (SVTMPC) is developed to overcome these limitations. Firstly, a zero steady-state error dynamic model is established, and a new update mechanism of the nominal state is devised to determine the unknown internal disturbance set of the AV system. Secondly, zonotopic representation of all defined sets is used to construct the prediction model, as well as a flexible tube with varying cross-sections is naturally designed to overcome excessive conservation and non-solution of Quadratic Programming (QP). Finally, a switching logic between conservative and radical strategies improves tracking performance under conventional conditions without compromising robust stability. Numerical simulation through three scenarios shows that the SVTMPC controller can comprehensively improve robust stability and adaptability compared with MPC and TRMPC. Hardware-in-the-Loop (HIL) experiment verifies the effectiveness and real-time of the SVTMPC controller.

Robust satisfaction of nonlinear performance constraints using barrier-based model predictive control

Efficient control of disturbed industrial systems requires methods to handle complex and nondifferentiable performance criteria given by customers directly in the control design process. In the design of control laws, our works evaluates nonlinear performance criterion for nonlinear systems subject to additive disturbances. Model Predictive Control using barrier functions is proposed. First of all, the stability of the method is proven in the linear case using Lyapunov function and invariant set theories. The presented law is also improved by considering robust tube-based Model Predictive Control for systems subject to additive disturbances. The method is then extended to nonlinear systems that neural networks can model when the knowledge-based model is unknown. The stability in the nonlinear case is not proven, but the method has shown its efficiency for different applications.

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.

Distributed Model Predictive Control for Connected and Automated Vehicles in the Presence of Uncertainty

Abstract This article focuses on the development of distributed robust model predictive control (MPC) methods for multiple connected and automated vehicles (CAVs) to ensure their safe operation in the presence of uncertainty. The proposed layered control framework includes reference trajectory generation, distributionally robust obstacle occupancy set computation, distributed state constraint set evaluation, data-driven linear model representation, and robust tube-based MPC design. To enable distributed operation among the CAVs, we present a method, which exploits sampling-based reference trajectory generation and distributed constraint set evaluation methods, that decouples the coupled collision avoidance constraint among the CAVs. This is followed by data-driven linear model representation of the nonlinear system to evaluate the convex equivalent of the nonlinear control problem. Finally, to ensure safe operation in the presence of uncertainty, this article employs a robust tube-based MPC method. For a multiple CAV lane change problem, simulation results show the efficacy of the proposed controller in terms of computational efficiency and the ability to generate safe and smooth CAV trajectories in a distributed fashion.

Data-Driven Synthesis of Robust Invariant Sets and Controllers

This paper presents a method to identify an uncertain linear time-invariant (LTI) prediction model for tube-based Robust Model Predictive Control (RMPC). The uncertain model is determined from a given state-input dataset by formulating and solving a Semidefinite Programming problem (SDP), that also determines a static linear feedback gain and corresponding invariant sets satisfying the inclusions required to guarantee recursive feasibility and stability of the RMPC scheme, while minimizing an identification criterion. As demonstrated through an example, the proposed concurrent approach provides less conservative invariant sets than a sequential approach.

A System Level Approach to Tube-Based Model Predictive Control

Robust tube-based model predictive control (MPC) methods address constraint satisfaction by leveraging an a priori determined tube controller in the prediction to tighten the constraints. This paper presents a system level tube-MPC (SLTMPC) method derived from the system level parameterization (SLP), which allows optimization over the tube controller online when solving the MPC problem, which can significantly reduce conservativeness. We derive the SLTMPC method by establishing an equivalence relation between a class of robust MPC methods and the SLP. Finally, we show that the SLTMPC formulation naturally arises from an extended SLP formulation and show its merits in a numerical example.

Towards Robust and High-performance Operations of Wave Energy Converters: an Adaptive Tube-based Model Predictive Control Approach

Model predictive control (MPC) is an effective control method to improve the energy conversion efficiency of wave energy converters (WECs). However, the current developed WEC MPC has not reached commercial viability since the control performance is significantly dependent on the WEC model fidelity. To overcome the plant-model mismatch issue in the WEC MPC control problem, this paper proposes a robust tube-based MPC method to bound plant states within disturbance invariant sets centered around the noise-free model trajectory. The invariant sets are also utilized for tightening the nominal model's constraints that robustly enable constraint satisfaction. Yet overly conservative invariant sets can narrow the feasible region of the states and control inputs, and hence a data-driven quantile recurrent neural network (QRNN) is proposed in this work to form a learning-based adaptive tube with reduced conservatism by quantifying WEC model uncertainties. The theoretical root is that time-dependent historical data can offer valuable insight into the future behaviour of uncertainties. Numerical simulations have validated that the proposed method can improve the energy capture rate compared to the TMPC approach, by synthesizing the QRNN-based tube with MPC.