Tube-based Robust Model Predictive Control Research Articles

110th Anniversary: Integration of Scheduling and Robust Model Predictive Control

In order to achieve optimal operational conditions, the integration of decision making across different layers of a company and the consideration of uncertain parameters in view of dynamic market conditions are essential. In this study, we propose a framework for the integration of scheduling and control for nonlinear systems under process uncertainties. The proposed approach includes the use of a tube-based robust model predictive control to handle disturbances affecting the control layer of the problem, the use of classification methodologies to determine the feasible space of operation of the process, and the use of surrogate models to derive the closed-loop input-output behavior of the dynamic system. Case studies are utilized to illustrate the performance of the proposed framework and evaluate the impact of control-level disturbances in scheduling solutions.

Robust Tube-Based Model Predictive Control for Lateral Path Tracking

This paper presents a path following application for vehicles based on a simple linear and time invariant single-track model, which is calculated based on a constant nominal longitudinal speed. In order to consider the differences in vehicle dynamics between the real vehicle and this constant nominal model, a tube-based robust Model Predictive Control (MPC) approach is introduced. The proposed control algorithm is designed in order to guarantee proper path tracking, not only considering lateral error, but also orientation error to the target trajectory. Additionally, strict constraints are considered in the control signal and the lateral path following error. The control approach is designed in order to guarantee certain comfort standards for a wide range of velocities, with guaranteed stability.

Tube-based robust output feedback model predictive control for autonomous rendezvous and docking with a tumbling target

In this paper, a tube-based robust output feedback model predictive control method (TRMPC) is proposed for controlling chaser spacecraft docking with a tumbling target in near-circular orbit. The controller contains a simple, stable, Luenberger state estimator and a tube-based robust model predictive controller. Several practical challenges are also considered under dock-enabling conditions, such as the control saturation, velocity constraint, approach corridor constraint, and collision avoidance constraint. Meanwhile, uncertainties are carefully analyzed when designing the controller, including dynamics uncertainty, measurement error, and control deviation. The TRMPC ensures that all possible state trajectories with uncertainties lie in the minimum robust positively invariant set (mRPI, i.e., the so-called tube in this paper). The tube center is the solution of a nominal (without uncertainties) system. Another important contribution of this paper is to propose a technique where it is unnecessary to calculate the mRPI explicitly. Thereby, the ‘curse of dimensionality’ can be avoided for a six-dimensional system. To verify the feasibility of the proposed TRMPC strategy in the presence of uncertainties, two scenarios of autonomous rendezvous and docking (AR&D) are simulated. The simulation results show that the TRMPC method is more efficient in minimizing the uncertainties, fuel consumption, and computational cost, compared to the classic model predictive control (MPC) method.

Attitude Control of a Small Spacecraft via Tube-Based Model Predictive Control

Precise attitude tracking control of small satellites is a demanding task due to the limited hardware resources of sensors, actuators, and processors equipped on board. Moreover, the effects of model uncertainty and persistent disturbance, such as gravity-gradient, magnetic, aerodynamic, and solar radiation pressure torques, on the satellite dynamics make the attitude control of miniaturized spacecraft more challenging. Consequently, under actuator limitations and stringent pointing requirements, robust controllers represent a more reliable solution to comply the mission constraints. In this paper, a tube-based robust model predictive control algorithm is exploited to address robust attitude control of CubeSat within an Earth-observation scenario, and its performance is evaluated. The proposed algorithm successfully achieves the three-axis stabilization required by the payload, using reaction wheels and applying constraints on the actuator torque and speed saturation as well as on pointing accuracy. To properly estimate the effectiveness of the proposed approach, the comparison with a nonlinear model predictive control is presented with the consideration of different sample times. The robustness of these two control schemes is then compared running 100 simulations, each one with random initial condition. The results demonstrate the effectiveness of the proposed robust algorithm for precise attitude tracking control of small satellites.

Trajectory Planning for Autonomous High-Speed Overtaking in Structured Environments Using Robust MPC

Automated vehicles are increasingly getting main-streamed and this has pushed development of systems for autonomous manoeuvring (e.g., lane-change, merge, and overtake) to the forefront. A novel framework for situational awareness and trajectory planning to perform autonomous overtaking in high-speed structured environments (e.g., highway and motorway) is presented in this paper. A combination of a potential field like function and reachability sets of a vehicle are used to identify safe zones on a road that the vehicle can navigate towards. These safe zones are provided to a tube-based robust model predictive controller as reference to generate feasible trajectories for combined lateral and longitudinal motion of a vehicle. The strengths of the proposed framework are: 1) it is free from non-convex collision avoidance constraints; 2) it ensures feasibility of trajectory even if decelerating or accelerating while performing lateral motion; and 3) it is real-time implementable. The ability of the proposed framework to plan feasible trajectories for high-speed overtaking is validated in a high-fidelity IPG CarMaker and Simulink co-simulation environment.

Tube-Based Active Robust MPC for Uncertain Constrained Linear Systems With Time Delays

The problem of robust tube-based model predictive control (MPC) is considered for a class of discrete-time constrained linear systems with time-delayed states and additional disturbances. This paper presents an active robust control scheme that can simultaneously cope with disturbances and time-delay states using active approaches, rather than inherent system robustness. The proposed robust controller design methodology is implemented by constructing a minimal robust control-invariant set of nominal linear systems with time-delayed states neglecting additional disturbances. Based on the time-delay-dependent robust MPC approach for the nominal time-delay system, a novel active robust tube-based MPC algorithm considering time-delayed states is proposed. Furthermore, a delay-dependent sufficient condition for the tube-based controller is derived. Based on the implementation of the delay-dependent robust tube-based controller, the novel active control method is much less conservative than previous approaches. Numerical simulation results demonstrate the effectiveness of the proposed method.

RMPC-Based Directional Stability Control for Electric Vehicles Subject to Tire Blowout on Curved Expressway

This paper presents a directional stability control based on robust tube-based model predictive control (RMPC) approach for an overactuated electric vehicle after tire blowout on curved expressway, in the presence of the exogenous disturbances, such as cross wind and road variation. To begin with, the vehicle dynamic simulation platform allowing for the tire vertical force redistribution after tire blowout is presented, and the reliability of the platform is further analyzed by comparing with the existing experimental test results. After that, a RMPC-based controller is designed to enhance the directional stability performance of the vehicle on curved expressway after tire burst. Also, a pseudo inverse switch control allocator is developed to realize the allocation of the desired resultant signal for the remained effective wheels at the last stage. In the end, the simulation results conducting on the depicted simulation platform demonstrate the favorable maneuverability of the proposed method over the conventional model predictive control (MPC) in enhancing directional stability performance of the vehicle after a tire blowout on curved expressway.

Tube-based robust model predictive control for spacecraft proximity operations in the presence of persistent disturbance

Rendezvous and Proximity Operations (RPOs) of two autonomous spacecraft have been extensively studied in the past years, taking into account both the strict requirements in terms of spacecraft dynamics variations and the limitations due to the actuation system. In this paper, two different Model Predictive Control (MPC) schemes have been considered to control the spacecraft during the final phase of the rendezvous maneuver in order to ensure mission constraints satisfaction for any modeled disturbance affecting the system. Classical MPC suitably balances stability and computational effort required for online implementation whereas Tube-based Robust MPC represents an appealing strategy to handle disturbances while ensuring robustness. For the robust scheme, the computational effort reduction is ensured adopting a time-varying control law where the feedback gain matrix is evaluated offline, applying a Linear Matrix Inequality approach to the state feedback stabilization criterion. An extensive verification campaign for the performance evaluation and comparison in terms of constraint satisfaction, fuel consumption and computational cost, i.e. CPU time, has been carried out on both a three degrees-of-freedom (DoF) orbital simulator and an experimental testbed composed by two Floating Spacecraft Simulators reproducing a quasi-frictionless motion. Main conclusions are drawn with respect to the mission expectations.

Robust Tube-Based Predictive Control for Visual Servoing of Constrained Differential-Drive Mobile Robots

This paper proposes a control strategy for the vision-based control problem of a nonholonomic constrained differential-drive mobile robots with bounded disturbance by using robust tube-based model predictive control (MPC) method. The proposed control strategy mainly consists of an ancillary state feedback controller and an MPC control for a nominal robotic system. First, the state-error kinematics of the nominal system is converted into a chained form system, and then its MPC optimization is computed to deal with a quadratic programming optimization problem by integrating a linear variable inequality-based primal-dual neural network (LVI-PDNN). Next, the gain scheduling of the ancillary state feedback is obtained via solving robust pole assignment using LVI-PDNN. An optimal state trajectory is thus generated for the nominal robotic system by the MPC without various uncertainties, the ancillary state feedback control forces the state variables to be constrained within an invariant designed tube. Finally, extensive experimental results on stabilization control of the nonholonomic mobile robot are provided to verify the effectiveness of the proposed robust tube-based MPC.

Tube-based MPC with Nonlinear Control for Load Transportation using a UAV

Abstract This paper presents a two-stage cascade control framework to solve hierarchically the trajectory tracking problem of a Tilt-rotor Unmanned Aerial Vehicle (UAV) carrying a suspended load. Initially, a nonlinear dynamic model is presented, which is after decoupled into two subsystems. The outer control system is designed by means of a robust tube-based Model Predictive Control (MPC) strategy, which is used to control the UAV’s planar motion and stabilize the suspended load. For the inner control system, the input-output feedback linearization (IOFL) technique combined with the dynamic extension approach and a discrete mixed H2/H∞ controller is considered to control the UAV’s altitude and attitude. Simulations results are carried out to corroborate the proposed control strategy.

Adaptive tube‐based model predictive control for linear systems with parametric uncertainty

A tube-based robust model predictive control (MPC) is proposed to be applied in constrained linear systems with parametric uncertainty. An estimation method is applied in this proposed technique to adapt the system model at each sampling time and to reduce the conservatism nature of the tube-based MPC as the system model approaches the real model as time passes. By updating the subject model online through this newly proposed approach the performance of the system is improved. Asymptotic stability of the closed-loop system is established. The simulation results of a DC motor are applied to illustrate the effectiveness of this proposed controller in dealing with one practical system.

Offset-Free Robust MPC of Systems with Mixed Stochastic and Deterministic Uncertainty

This paper presents a robust model predictive control (MPC) approach for offset-free tracking of piece-wise constant references in the presence of bounded deterministic and stochastic disturbances. The system is considered to be linear with two sources of additive bounded uncertainties on the states. The first uncertainty source accounts for unknown, deterministic structural/parametric plant-model mismatch. The second uncertainty source represents stochastic exogenous system disturbances. The proposed deterministic-stochastic robust MPC approach uses estimates of the deterministic model uncertainties to modify the nominal state and input targets. This allows for achieving offset-free tracking of the mean of the controlled variables. A non-conservative constraint tightening procedure is used to handle probabilistic state constraints and hard input constraints in the presence of stochastic uncertainties. The computational complexity of the proposed robust MPC approach is comparable to that of nominal MPC. The closed-loop performance of the proposed robust MPC approach is compared to that of robust tube-based MPC and stochastic MPC in a simulation study.

Output feedback polytopic LPV Tube-RMPC control for air-breathing hypersonic vehicles

This paper proposes an output feedback tube-based robust model predictive control (Tube-RMPC) scheme for air-breathing hypersonic vehicles. Firstly, the polytopic linear parameter varying (LPV) model is established by Jacobian linearisation and tensor-product (T-P) modelling for the longitudinal dynamics. Secondly, a robust polytopic state observer is designed to estimate the attack angle, pitch angle and pitch rate by utilising the limited information of altitude and velocity. Then, the Tube-RMPC controller satisfying all vertices of polytopic LPV model is presented. Wherein, estimation error and control error are bounded by two robust positively invariant sets respectively. The Tube-RMPC controller ensures that actual states lie in a tube centred with the nominal states. The input constraints of original system are guaranteed by employing tighter constraints for nominal system. Finally, simulation results show the feasibility and effectiveness of the proposed method for hypersonic vehicles.

Multi-Level Predictive Control for Energy Management of Hybrid Ships in the Presence of Uncertainty and Environmental Disturbances**This research is supported by the project ShipDrive: A Novel Methodology for Integrated Modelling, Control, and Optimization of Hybrid Ship Systems (project 13276) of the Dutch Technology Foundation STW.

Abstract: In this paper, a control methodology is proposed to address the problem of energy management in hybrid ships in the presence of disturbances applied by the surrounding environment and taking the uncertainties within the ship model into account. Nonlinear Robust Tube-Based Model Predictive Control (NRTB-MPC) is adopted to control the ship speed and calculate the demanded energy for propelling the ship. At the same time, a conventional MPC approach is proposed to track the demanded energy and determine the optimal split between different energy sources. Simulation experiments using a hybrid ship system illustrate the potential of the approach.

An Off-Line Formulation of Tube-Based Robust MPC Using Polyhedral Invariant Sets

In this paper, an off-line formulation of tube-based robust model predictive control (MPC) using polyhedral invariant sets is proposed. A novel feature is the fact that no optimal control problem needs to be solved at each sampling time. Moreover, the proposed tube-based robust MPC algorithm can deal with the linear time-varying (LTV) system with bounded disturbance. The simulation results show that the state at each time step is restricted to lie within a tube whose center is the state of the nominal LTV system that converges to the origin. Finally, the state is kept within a tube whose center is at the origin, so robust stability is guaranteed. Satisfaction of the state and control constraints is guaranteed by employing tighter constraint sets for the nominal LTV system.

A computationally efficient robust tube based MPC for linear switched systems

This article considers the robust regulation problem for a class of constrained linear switched systems with bounded additive disturbances. The proposed solution extends the existing robust tube based model predictive control (RTBMPC) strategy for non-switched linear systems to switched systems. RTBMPC utilizes nominal model predictions, together with tightened sets constraints, to obtain a control policy that guarantees robust stabilization of the dynamic systems in presence of bounded uncertainties. In this work, similar to RTBMPC for non-switched systems, a disturbance rejection proportional controller is used to ensure that the closed loop trajectories of the switched linear system are bounded in a tube centered on the nominal system trajectories. To account for the uncertainty related to all sub-systems, the gain of this controller is chosen to simultaneously stabilize all switching dynamics. The switched system RTBMPC requires an on-line solution of a Mixed Integer Program (MIP), which is computationally expensive. To reduce the complexity of the MIP, a sub-optimal design with respect to the previous formulation is also proposed that uses the notion of a pre-terminal set in addition to the usual terminal set to ensure stability. The RTBMPC design with the pre-terminal set aids in determining the trade-off between the complexity of the control algorithm with the performance of the closed-loop system while ensuring robust stability. Simulation examples, including a Three-tank benchmark case study, are presented to illustrate features of the proposed MPC.

Tube-based robust economic model predictive control

In this paper, we develop a tube-based economic MPC framework for nonlinear systems subject to unknown but bounded disturbances. Instead of simply transferring the design procedure of tube-based stabilizing MPC to an economic MPC framework, we rather propose to consider the influence of the disturbance explicitly within the design of the MPC controller, which can lead to an improved closed-loop average performance. This will be done by using a specifically defined integral stage cost, which is the key feature of our proposed robust economic MPC algorithm. Furthermore, we show that the algorithm enjoys similar properties as a nominal economic MPC algorithm (i.e., without disturbances), in particular with respect to bounds on the asymptotic average performance of the resulting closed-loop system, as well as stability and optimal steady-state operation.

Robust Time-Varying Model Predictive Control with Application to Mobile Robot Unmanned Path Tracking

This work presents a tube-based Robust Model Predictive Controller (RMPC) with an application to the control of a mobile robot performing unmanned path tracking. In addition to robustness against unknown but bounded uncertainties, the RMPC algorithm proposed in this work is implementable (i.e. it imposes low complexity and conservativeness), thus making it highly amenable to real-time applications. This is due to an improved formulation of the tube over previous works (here a tube is sequence of sets) which is constructed by predicting the evolution of the difference between the actual uncertain system under control and its respective nominal disturbance-free system. Moreover, a feedback corrective controller formulated as a time-varying finite-time Linear Quadratic Regulator (LQR) is proposed to regulate the uncertain system around its respective nominal uncertainty-free system and thereby suppress the effect of uncertainties acting on the actual system. The proposed RMPC algorithm is applied to a Pioneer P3 –DX mobile robot platform performing unmanned path tracking. Experimental results demonstrate robust and stable performance of the proposed RMPC algorithm.

A dual-terminal set based robust tube MPC for switched systems

This article considers the robust regulation problem for a class of constrained linear switched systems with bounded additive disturbances. The proposed solution extends the existing robust tube based model predictive control (RTBMPC) strategy for non-switched linear systems to switched systems. RTBMPC utilizes nominal model predictions, together with tightened sets of state and input constraints, to obtain a control policy that guarantees robust stabilization of the dynamic systems in presence of bounded uncertainties. Similar to RTBMPC for non-switched systems, a disturbance rejection proportional controller is used to ensure that the closed loop trajectories of the switched linear system are bounded in a tube centered on the nominal system. To account for the switching dynamics, the gain of this controller is chosen to simultaneously stabilize all switching dynamics. The RTBMPC for the switched system requires an on-line solution of a Mixed Integer Quadratic Program (MIQP). To reduce the complexity of the MIQP, a sub-optimal design is proposed, which considers the notion of a pre-terminal set in addition to the usual terminal set used to ensure stability. The RTBMPC design with the pre-terminal set aids in tuning the trade-off between the complexity of the control algorithm with the optimal performance of the closed-loop system while ensuring robust stability. Examples are presented to illustrate features of the proposed MPC.

On the explicit dead-time compensation for robust model predictive control

Computational simplicity is one of the most important aspects to take into account in robust model predictive control (MPC). In dead-time processes, it is common to use an augmented state-space representation in order to apply robust MPC strategies but, this procedure may affect computational aspects. In this paper, explicit dead-time compensation will be used to avoid augmented representation. This technique will be analyzed in terms of robust stability and constraint satisfaction for discrete-time linear systems. The results of this discussion will be applied to a robust tube-based MPC strategy which is able to guarantee robust stability and constraint satisfaction of a dead-time system by considering a prediction model without dead-time. Moreover, taking advantage of the proposed scheme, the robust MPC will be particularized for first-order plus dead-time models which simplifies significantly controller synthesis. The proposed dead-time compensation method will be applied to different robust MPC strategies in two case studies: (i) a simulated quadruple-tank system, and (ii) an experimental scaled laboratory heater process.