Tube-based Model Predictive Control Research Articles

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%.

Autonomous Spacecraft Rendezvous Using Tube-Based Model Predictive Control: Design and Application

As the concentration of large space debris increases, how rendezvous maneuvers involving these typically non-cooperative, freely tumbling bodies are planned and executed is evolving. The rendezvous must be carefully planned, employing up-to-date in situ data to identify the inertial and motion parameters of the target body, and executed in a manner that accounts for the remaining uncertainty in these parameters. This paper presents an extension of the Tumbling Rendezvous via Autonomous Characterization and Execution (TRACE) pipeline used in the ROAM/TumbleDock Astrobee experiment campaign, which sequences the target state estimation, motion planning, controller design, and maneuver execution tasks while additionally providing logical loop-back avenues to previous tasks, increasing the chances of a successful maneuver. The pipeline’s performance is analyzed in simulation, utilizing target state estimates generated in a previous activity on a dedicated on-ground test bed; online motion planning, based on nonlinear programming and warm-started using a trajectory library generated offline with a novel graphics-processing-unit-based method; and tube-based model predictive control to robustly track the planned trajectory. Tube-based model predictive control is an actively evolving subject, distributed over multiple publications and various research interests. The necessary theory and considerations for practical implementation of the method are consolidated; its use, features, and limitations in the proposed task are demonstrated.

Asynchronous Computation of Tube-based Model Predictive Control

Tube-based model predictive control (MPC) methods bound deviations from a nominal trajectory due to uncertainties in order to ensure constraint satisfaction. While techniques that compute the tubes online reduce conservativeness and increase performance, they suffer from high and potentially prohibitive computational complexity. This paper presents an asynchronous computation mechanism for system level tube-MPC (SLTMPC), a recently proposed tube-based MPC method which optimizes over both the nominal trajectory and the tubes. Computations are split into a primary and a secondary process, computing the nominal trajectory and the tubes, respectively. This enables running the primary process at a high frequency and moving the computationally complex tube computations to the secondary process. We show that the secondary process can continuously update the tubes, while retaining recursive feasibility of the primary process.

Tube-based MPC of hierarchical distribution network for voltage regulation considering uncertainties of renewable energy

• A novel model for low-voltage distribution areas based on genetic algorithm is developed, which simplifies the complexity of online computation and suppresses the inner voltage deviations in a safe section. • The coordination of PVs and ESs at different voltage levels is promoted to improve the voltage regulation while short of the reactive power capacity of PVs. • A tube-based MPC controller is proposed for voltage fluctuation under uncertainties introduced by power disturbance and a genetic-algorithm-based equivalent model of low-voltage areas, with which a distribution network can work in common conditions under variable operations of PVs. With the ever-growing penetration of distributed generators (DGs), voltage deviation caused by randomness and fluctuation has become increasingly prominent in distribution networks. Conventional voltage control cannot sufficiently utilize the rapid and flexible characteristics of DGs. In this paper, a tube-based model predictive control (TMPC) method is proposed to address voltage violation and fluctuation difficulties considering the uncertainties of renewable energy. Firstly, a simplified model approach based on a genetic algorithm (GA) is proposed to unify numerous DGs in low-voltage distribution areas. Then, based on this simplified model combined with the voltage sensitivity model, the TMPC controller is designed for the whole distribution network, which consists of two subsidiary controllers to suppress voltage deviation caused by uncertainties. The former is used to generate a central path for different operating points of the grid, and the latter is designed to track the path in a bound tube. The proposed method can cope with the uncertainties caused by external disturbances such as drastic power fluctuations. Finally, the numerical simulation in the case studies demonstrates the effectiveness of the proposed method in accuracy and efficiency.

Tube-based coalitional MPC with plug-and-play features

This paper presents a distributed setting of model predictive control (MPC) to manage linear multi-agent systems consisting of coupled subsystems. Specifically, local controllers can work in coalitions to improve performance and handle plug-and-play events. This study provides insight into a coalitional MPC strategy based on optimized tubes that handles plug-in and plug-out subsystems. Moreover, we explore an inherent robustness gap to absorb disturbances not covered by the tubes without having to group local controllers. A comparison of our approach with centralized and decentralized MPC is reported using an illustrative example.

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 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.

Contraction-based model predictive control for stochastic nonlinear discrete-time systems with time-varying delays via multi-dimensional Taylor network

For stochastic nonlinear systems with time-varying delays, the existing robust control approaches are unnecessarily conservative in most practical scenarios. Within this context, a mathematically rigorous and computationally tractable tube-based model predictive control scheme is proposed in the framework of contraction theory. A contraction metric is systematically constructed via convex optimization by forming a differential LyapunovKrasovskii function on tangent space. It guarantees the perturbed actual solution trajectories to be contained within a robust positive invariant tube centered along the reference trajectories and results in an explicit exponential bound on the deviation. The application scenarios of the control contraction metric controller are extended from constant delay systems into time-varying delay systems thereby. Compared with the existing robust mechanism for time-delay systems based on min-max optimization formulation with a linear feedback controller, the proposed scheme greatly reduces the design conservativeness and yields a larger region of attraction. A sparse multi-dimensional Taylor network (MTN) is designed to parameterize the family of the geodesic. Compared to conventional NNs and MTN surrogates, sparse MTN features a more concise topology that enhances its computational efficiency conspicuously. Results of the numerical simulations verify the effectiveness of the proposed method.

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.

Fault-Tolerant Control for Path-Following of Independently Actuated Autonomous Vehicles Using Tube-Based Model Predictive Control

This paper designs a fault-tolerant controller for the path-following of independently actuated (IA) electric autonomous vehicles (AVs) with steer-by-wire (SBW) systems. Such a controller can handle the effect of steering motor fault and the bounded disturbances of the vehicle dynamic system. Firstly, we formulate the dynamic models of the vehicle, the tire, the SBW system, and the steering motor. Secondly, with the models, we identify the effectiveness coefficient of the steering motor and calculate the steering resistance torque. Both the two parameters are used to determine the potential maximum front-wheel angle. The effectiveness coefficient can determine when the control system switches to the fault-tolerant control (FTC) mode. For the FTC mode, we then introduce a tube-based model predictive control (MPC) framework to guarantee vehicle stability in steering processes and maintain the tracking performance. The disturbances of the vehicle dynamic system affect both the control input and the state. On one hand, the disturbances are formulated to be a tightened state constraint. On the other hand, together with the potential maximum front-wheel angle, the disturbances are formulated to be a tightened input constraint. Finally, we show the effectiveness of the designed fault-tolerant controller via Carsim-Simulink joint simulation and real-vehicle experiment.

Event-triggered robust model predictive control with stochastic event verification

This paper is concerned with event-triggered robust model predictive control (MPC) for linear discrete-time systems with bounded disturbances. Based on the ergodicity of a purposely designed Markov chain, a stochastic triggering scheme involving a prescribed triggering function, an updating law for the transition probabilities of the Markov chain, and a checking function is proposed to achieve aperiodic and non-persistent event verification and enlarge the inter-execution time. Both tube-based MPC and linear matrix inequality-based (LMI-based) MPC are considered, and they show complementary merits with such a stochastic triggering scheme. Under mild conditions, recursive feasibility and closed-loop robust stability of both approaches are guaranteed theoretically. Simulation results are provided to show the effectiveness and merits of the proposed approaches.

Tube-Based Model Predictive Control with Uncertainty Identification for Autonomous Spacecraft Maneuvers

Autonomous spacecraft must be robust toward uncertainty and disturbances in the system while achieving required levels of performance. As an example, robotic servicing spacecraft must intercept target objects with potentially unknown dynamic properties during active debris removal efforts. This work presents tube-based model predictive control (MPC) with uncertainty identification as a strategy to enhance performance while maintaining robustness in autonomous maneuvers. The proposed algorithm, which is an extension of the standard tube-based MPC framework, measures and predicts the exogenous input to the system (i.e., the uncertainty) online. This in turn enables the robust tube to be shrunk and grown appropriately as the trajectory progresses in both predicted time and actual time. The algorithm is demonstrated in a simulated intercept maneuver with a tumbling target whose inertia tensor is uncertain. Results indicate two improvements over the standard tube-based algorithm: first, better performance is obtained when the initial exogenous input bounds are overconservative, and, second, there is greater flexibility in encouraging robustness when the exogenous input bounds are overly optimistic since the robust tube is updated online. As such, tube-based MPC with uncertainty identification represents an incremental step in enhancing the flexibility of autonomous spacecraft in addressing uncertain scenarios and environments.

Learning for MPC with stability & safety guarantees

The combination of learning methods with Model Predictive Control (MPC) has attracted a significant amount of attention in the recent literature. The hope of this combination is to reduce the reliance of MPC schemes on accurate models, and to tap into the fast developing machine learning and reinforcement learning tools to exploit the growing amount of data available for many systems. In particular, the combination of reinforcement learning and MPC has been proposed as a viable and theoretically justified approach to introduce explainable, safe and stable policies in reinforcement learning. However, a formal theory detailing how the safety and stability of an MPC-based policy can be maintained through the parameter updates delivered by the learning tools is still lacking. This paper addresses this gap. The theory is developed for the generic robust MPC case, and applied in simulation in the robust tube-based linear MPC case, where the theory is fairly easy to deploy in practice. The paper focuses on reinforcement learning as a learning tool, but it applies to any learning method that updates the MPC parameters online.

Tube-Based Adaptive Model Predictive Control for Deviation Correction in Vertical Drilling Process

The build-up rate of bottom hole assembly (BHA) fluctuates violently during deviation correction in vertical drilling process due to the mutation of stratum, low operation accuracy of BHA, etc. As a result, the uncertainty of the build-up rate of BHA has great influence on the performance of correction control and the quality of drilling trajectory. To deal with this control problem, a tube-based adaptive model predictive control (MPC) method is developed in this paper. By dividing the trajectory extension model into two lower-order models, the calculability of the disturbance invariant set of the tube-based MPC controller is ensured. Meanwhile, the bootstrap sampling method with a sliding window is used to estimate the range of the uncertainty online to reduce the conservativeness of the controller based on practical drilling data. Moreover, rules and a small interim deviation are introduced to adjust the parameters adaptively to decrease the curvature of the drilling trajectory and the calculation time. Simulations and hardware-in-the-loop experiment tests are carried out to illustrate the effectiveness of the control method. The results show that the proposed method completes the drilling task with higher quality of trajectory and needs less time for calculation.

Adaptive Model Predictive Control-Based Path Following Control for Four-Wheel Independent Drive Automated Vehicles

Due to inevitable parameter uncertainties and disturbances, four-wheel independent drive automated vehicles (4WIDAVs) will produce tracking deviation during the path following process, which have a negative impact on driving safety. Meanwhile, the over-actuated feature of 4WIDAVs will also increase the deviation if not properly handled. To solve this problem, a specific adaptive model predictive control strategy for path following of 4WIDAVs is proposed. Firstly, to obtain a real-time and accurate vehicle dynamics model, the recursive least square method is used to estimate the time-varying uncertainty of tire cornering stiffness. Secondly, based on the real-time updating system model, the modified tube-based model predictive control method is applied to realize path following under the influence of the disturbance. Meanwhile, the compensating yaw moment for controlling vehicle is generated by the designed torque distribution algorithm, which makes full use of the over-actuated feature of 4WIDAVs. Finally, different maneuvers are performed both in simulation and experiment. Results show that the proposed strategy can achieve more accurate path following than the traditional model predictive control and linear quadratic regulator. Compared with the existing controller, the path following accuracy is improved by 41.6% and 60% in simulation and experiment, respectively. Therefore, the proposed strategy is proved to be effective, which provides a theoretical reference for vehicle control in reality.

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.

System level disturbance reachable sets and their application to tube-based MPC

Tube-based model predictive control (MPC) methods leverage tubes to bound deviations from a nominal trajectory due to uncertainties in order to ensure constraint satisfaction. This paper presents a novel tube‑based MPC formulation based on system level disturbance reachable sets (SL‑DRS), which leverage the affine system level parameterization (SLP). We show that imposing a finite impulse response (FIR) constraint on the affine SLP guarantees containment of all future deviations in a finite sequence of SL‑DRS. This allows us to formulate a system level tube‑MPC (SLTMPC) method using the SL‑DRS as tubes, which enables concurrent optimization of the nominal trajectory and the tubes, while using a positively invariant terminal set. Finally, we show that the SL‑DRS tubes can also be computed offline.

A hierarchical control scheme for multiple aerial vehicle transportation systems with uncertainties and state/input constraints

Multiple aerial vehicles have the potential to transport payload in any direction and with any orientation in Special Euclidean Group. In this article, the control problem of the multiple aerial vehicle transportation system with uncertainties and state/input constraints is considered. The multiple aerial vehicles connect with the load via spherical pairs that coincide with the center of mass of the aerial vehicles. A hierarchical control scheme of such challenging complex systems is proposed. The outer loop is designed by a tube-based model predictive control in order to deal with uncertainties, state constraints and input boundedness, while the inner loop is designed by robust control technique which forces the attitude tracking error of the aircraft into robust invariant set. The attitude tracking error of the inner loop induces a difference between the commanded and the actual equivalent wrench acting on the load. Such difference is treated as the equivalent disturbance of the outer loop to guarantee the fulfillment of the state and input constraints. The hierarchical control structure simplifies the design procedure, while still preserves the convergence and feasibility of the overall controlled system under uncertainties, which are proved strictly in the paper. Numerical simulations on a six-quadrotor transportation system are conducted to demonstrate the performance of the proposed control scheme. A real-world prototype including three quadrotors is developed. The hardware-in-the-loop simulation on the prototype supports the real-time feasibility of the proposed scheme.

Robust event-triggered model predictive control of polytopic LPV systems: An input-to-state stability approach

This paper addresses the problem of designing robust event-triggered tube-based model predictive control (TMPC) for constrained linear parameter-varying systems. The main challenge with implementing TMPC schemes in an event-triggered setup is that the assumption of feedback for all sampling times is not satisfied, which leads to an exponential growth of the predicted sets. First, a novel modification of homothetic TMPC is introduced to consider the effect of open-loop control between triggering times. At each sampling instant, the TMPC provides the maximal feasible number of open-loop steps and a sequence of control inputs, which can be applied in an open-loop fashion while satisfying the state and control constraints. Second, depending on the maximal number of open-loop steps, the triggering condition is derived based on the input-to-state stability (ISS) concept with the aim to reduce the communication traffic between sensors, controllers and actuators. Moreover, recursive feasibility and asymptotic stability of the closed-loop system are guaranteed. Simulation results demonstrate the efficacy of the proposed scheme while reducing the frequency of the update times.

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.