Robust Model Predictive Control Approach Research Articles

Simultaneous robust model predictive control and state estimation for nonlinear systems with state‐ and input‐dependent uncertainties

AbstractThe convergence and stability of uncertain nonlinear systems is a challenging problem in the nonlinear control area. Besides, in many practical cases, all states are not measurable and are affected by measurement noise. Based on this motivation, the first objective of this paper is to design a novel output feedback robust model predictive control approach for nonlinear systems with state‐ and input‐dependent uncertainties and measurement noise. This approach combines state estimation and robust model predictive control (MPC) into one min–max optimization and by solving the optimization, these two tasks are performed simultaneously. The studied nonlinear system comprises a linear part, a nonlinear part, and a function that denotes the state‐ and input‐dependent uncertainties. Therefore, the other objective is to reduce the computational complexity; thus, the system's nonlinear term and the aforementioned uncertainties are converted into additional disturbances. Subsequently, the optimization problem becomes a quadratic form, which leads to global convergence with the appropriate selection of objective function weights. Besides, this paper explores the convergence of the closed‐loop system states and the sufficient synthesis conditions to guarantee input‐to‐state stability. The implementation on a numerical example and a CSTR process demonstrate the applicability and reliability of the proposed approach.

Robust H∞ observer-based model predictive controller for uncertain linear discrete-time systems due to external disturbances

This article investigates an observer-based robust model predictive control (RMPC) design to control the uncertain discrete-time linear systems with disturbances. To make a more practical scheme, it is supposed that the uncertain system has been faced with unknown disturbance and input constraints. The proposed RMPC approach is based on a state feedback control design that ensures the [Formula: see text] performance criterion to attenuate disturbance affections. Furthermore, in view of practical application, the control law is constructed based on the estimated states obtained from the Luenberger observer. Based on Lyapunov’s theory, the input to state practically stability (ISPS) of the closed-loop system is ensured. Appropriate conditions for the ISPS of the closed-loop system and the estimation error are obtained in terms of online linear matrix inequalities (LMIs) which lead to obtaining the time-varying gain matrices of both controller and observer. Finally, to validate the obtained results, the proposed approach is applied to a numerical example and it is compared with the existing control scheme and the superiority is proved.

Single-Loop Robust Model Predictive Speed Regulation of PMSM Based on Exogenous Signal Preview

Speed regulation problem of permanent magnet synchronous motor (PMSM) with overcurrent protection is addressed in this paper by using a single-loop robust model predictive control (MPC) approach. The unique feature of this new disturbance rejection MPC method is the utilization of the preview information of lumped exogenous signal. Different from most of the existing disturbance observer-based MPC approaches, the reference inputs, the lumped disturbances in PMSM, and their higher-order differences are explicitly considered as a part of this new concept of lumped exogenous signal by exploiting output regulation theory. Preview of the possible future behaviors of this exogenous signal is conducted based upon the development of two generalized proportional integral observers (GPIOs). An accuracy enhanced prediction model is obtained by exploiting the estimation and preview of the exogenous signal during the prediction horizon. Speed regulation problem of the PMSM is transformed into an error stabilization problem by means of regulator equations and preview information where the faster time-varying exogenous signal is considered. The overcurrent protection problem is also taken into account throughout the receding horizon optimization process. Experimental studies show that the proposed method achieves effective functionality in overcurrent protection, and much better dynamic performance against time-varying exogenous inputs.

Robust constraint satisfaction and stability of virtually coupled train set with uncertain dynamics: A dual-mode robust MPC approach

As a hot topic in railways, virtual coupling technology connects multiple trains (units) as a virtually coupled train set (VCTS) and aims to improve the performance of train operation by keeping a small following distance between successive units. For the desired performance, robust constraint satisfaction and stability are crucial properties of a VCTS. However, they are hard to be simultaneously guaranteed with uncertain train dynamics. To solve this problem, we propose a dual-mode robust model predictive control (DRMPC) approach. First, by considering uncertain actuator gains and resistance parameters, the VCTS operation is modeled with safety and input constraints. Then, we put forward a dual-mode scheme to adopt a controller from two candidates. Among this, we formulate a feedback controller through the analysis of local and string stability and adopt it in an admissible region to efficiently guarantee the crucial properties. To handle the rest cases with the risk of constraint dissatisfaction, an optimal controller is yielded by solving a constrained optimization framework employing the homothetic-tube technique. Specifically, local tubes are recursively constructed online by the set-membership prediction, while considering the tubes from the preceding unit. Next, the robust constraint satisfaction and stability of the VCTS are proved within the closed-loop implementation of DRMPC. Finally, experimental results verify the performance and advantages of the proposed approach on a VCTS.

A robust MPC approach for platooning control of automated vehicles with constraints on acceleration

This paper deals with the issue of robust model predictive control (MPC)-based automated vehicle platooning control under acceleration constraints. A unified framework is established to characterize the dynamics of cooperative longitudinal motion of the vehicle platoon with parameter uncertainties as well as the saturated control input. More specifically, the parameter uncertainties are described by norm bounded, and the saturated control input is assumed to be sector-bounded. The coordinated platoon behaves like a linear model where the deviations from the reference spacing and velocity are selected as the state variables. The main objective of this paper is to develop a robust MPC-based platooning control such that the controlled vehicular platoon can track the leading vehicle under the acceleration constraint and parameter uncertainties, meanwhile maintaining safety and resource management. By solving the optimization problem, sufficient conditions for MPC-based platooning control system stability in the mean-squared sense and performance analysis are provided and the controller gain is obtained at each instant. Finally, multi-vehicle cases are conducted to demonstrate the feasibility of the developed control algorithm.

Design and Verification of Offline Robust Model Predictive Controller for Wheel Slip Control in ABS Brakes

Wheel slip control is a critical aspect of vehicle safety systems, notably the antilock braking system (ABS). Designing a robust controller for the ABS faces the challenge of accommodating its strong nonlinear behavior across varying road conditions and parameters. To ensure optimal performance during braking and prevent skidding or lock-up, the ideal wheel slip value can be determined from the peak of the tire–road friction curve and maintained throughout the braking process. Among various control approaches, model predictive control (MPC) demonstrates superior performance and robustness. However, online MPC implementation encounters significant computational burdens and real-time limitations, particularly when dealing with larger problem sizes. To address these issues, this study introduces an offline robust model predictive control (RMPC) methodology. The proposed approach is based on the robust asymptotically stable invariant ellipsoid methodology, which employs linear matrix inequalities (LMIs) to calculate a collection of invariant state feedback laws associated with a sequence of nested invariant stable ellipsoids. Simulation results indicate a significant reduction in computational burden with the offline RMPC approach compared to online implementation, while effectively tracking the desired wheel slip reference values and system constraints. Moreover, the offline RMPC design aligns well with the online MPC design and verifies its effectiveness in practice.

Ellipsoidal tube‐based output feedback robust MPC for linear systems with bounded disturbances and noises

AbstractThis paper designs an ellipsoidal tube‐based output feedback robust model predictive control approach for linear systems with bounded disturbances and noises subject to physical constraints. The ellipsoidal tube‐based output feedback robust model predictive control approach combines the off‐line optimization to compute controller parameters and deal with system uncertainties, and the on‐line optimization to stabilize the nominal system with time‐varying tightened constraints. In the off‐line optimization, the state observer gain, ancillary feedback controller gain, and tightened constraint sets on the nominal system are synthesized in one optimization. Then, an ellipsoidal terminal constraint set with terminal controller gain and terminal cost matrix, and tightened constraint sets related to different sizes of estimation error bounds are computed. In the on‐line model predictive control optimization with time‐varying tightened constraints, a sequence of nominal control inputs is receding horizon optimized to steer the nominal system state to the ellipsoidal terminal constraint set. When the current nominal system state is bounded within the ellipsoidal terminal constraint set, the nominal control inputs are switched to the feedbacks on the closed‐loop nominal system states. Recursive feasibility of the proposed algorithm and robust stability of the controlled system are guaranteed.

Sufficient Conditions for Convergent Recursive Extrapolation of qLPV Scheduling Parameters Along a Prediction Horizon

Model Predictive Control (MPC) algorithms have long been applied to nonlinear processes. In a quasi-Linear Parameter Varying (qLPV) setting, nonlinearities are included into bounded scheduling parameters, which are given as a function of endogenous variables; these scheduling parameters are <i>a-priori</i> unknown along a future prediction horizon, which complicates MPC design. To address this problem, literature points out two options: robust MPC approaches, considering the scheduling to be uncertain; or sub-optimal ones, that set values for these parameters along the horizon. With respect to the latter group, this paper proposes an extrapolation algorithm that estimates the future values of the qLPV scheduling parameters for a fixed prediction horizon of <inline-formula><tex-math notation="LaTeX">$N$</tex-math></inline-formula> steps; the method is based on a recursive procedure using simple Taylor expansions. Sufficient conditions for convergent extrapolation are presented with regard to the form and class of the scheduling function and the robustness of the MPC. Different benchmark examples from the literature are presented to illustrate the algorithm, which is also compared to state-of-the-art techniques.

A robust MPC approach with controller tuning for close following operation of virtually coupled train set

Virtual coupling, as an emerging concept in railways, expects successive trains in a virtually coupled train set (VCTS) to maintain a short following distance. However, this close following operation is still difficult to be achieved given uncertain resistances and nonlinear safety constraints. To solve this problem, this paper proposes a robust model predictive control (RMPC) approach for the close following operation of VCTS while satisfying a nonlinear safety constraint with relative braking principle. First, we construct a robust positively invariant set that bounds the tracking errors caused by uncertain resistances. Further, a semi-definite program-based controller tuning algorithm is proposed to reduce the following distance in the premise of the tightened constraint for robustly satisfying the nonlinear safety constraint. Then, by mathematically examining the future trajectories of successive trains, we create a terminal constraint set to ensure the recursive feasibility of the proposed RMPC. This closed-loop property guarantees the satisfaction of the safety constraint in any situation, even in the case of sudden deceleration of VCTS. Finally, numerical experiments are conducted to evaluate the following distance with respect to heterogeneous trains and verify the effectiveness of our approach. Experimental results demonstrate that the expected close following operation can be achieved while robustly satisfying the nonlinear safety constraint with uncertain resistances. Moreover, our approach further reduces the following distance in a VCTS by over 5%, compared with existing research.

Optimal control of inventory level for perishable goods with uncertain decay factor and uncertain forecast information: a new robust MPC approach

This paper deals with the inventory control in supply chains under the following assumptions: (1) perishable goods with uncertain deteriorating factor, (2) a future uncertain customer demand that, over a limited prediction horizon, belongs to a known compact set. The problem is to define a smooth control policy maximising the fulfilled customer demand and minimising the inventory level. This problem is here solved through a new Robust Model Predictive Control (RMPC) approach. This implies solving a min–max optimisation problem with hard constraints on the control effort (i.e. the sequence of replenishment orders). To drastically reduce the numerical complexity of this problem, the control signal is sought in the space of B-spline functions, which are known to be universal approximators admitting a parsimonious parametric representation. This allows us: (1) to reduce the number of both decision variables and constraints involved in the optimisation procedure, (2) to reformulate the numerically demanding minimisation of the worst case cost functional as a simpler Weighted Constrained Robust Least Squares (WCRLS) estimation problem. The WCRLS algorithm can be efficiently solved using interior point methods. A rigorous analysis of stability and feasibility conditions is provided.

Robust coalitional model predictive control with negotiation of mutual interactions

This article presents a robust coalitional model predictive control (MPC) approach where neighboring agents negotiate the bounds of their coupling variables. Also, the control variables of each agent are divided into a private part that is locally optimized, and a public part that is controlled by the corresponding neighbors. Under certain conditions, the agents communicate to update the bounds that determine the constraints on these private and public variables. Moreover, the mutual disturbances induced by coupling are considered using a tube-based approach, guaranteeing recursive feasibility and stability of the closed-loop system. The proposed method is tested on a simulated eight-input coupled tank benchmark to illustrate its benefits.

Dual‐mode robust model predictive control for the tracking control of nonholonomic mobile robot

AbstractIn this article, a novel dual‐mode robust model predictive control (MPC) approach is proposed for solving the tracking control problem of nonholonomic mobile robot with additive bounded disturbance. To reduce the negative effects of disturbance, and drive the state of real system closer to the one of nominal system, a robust reference signal is introduced into the cost function of MPC. In order to reduce the computation burden caused by online optimization of MPC and further improve the tracking accuracy, a dual‐mode control strucuture consisting of the robust MPC and the local nonlinear robust control is developed, in which the local nonlinear robust control law is applied within a specified terminal region. Finally, simulation results on the nonholonomic mobile robot are presented to show the validity of the proposed control approach.

Designing a resilient supply chain through a robust adaptive model predictive control policy under perishable goods and uncertain forecast information

We consider the inventory control problem for Supply Chains (SC) with deteriorating items and an uncertain future customer demand freely varying inside a given compact set. The problem is to define a resilient Replenishment Policy (RP) keeping the actual inventory level as close as possible to a desired reference trajectory. This requirement should be satisfied despite uncertainties on the decay factor of stocked goods and unexpected customer demand patterns outside the bounds of the compact set. We propose a method based on a Resilient Robust Model Predictive Control (RRMPC) approach. This requires dealing with a Min-Max Constrained Optimization Problem (MMCOP). To reduce the numerical complexity of the algorithm, the control signal is parametrized using B-spline functions.

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.

Robust Optimal Control for Demand Side Management of Multi-Carrier Microgrids

This paper focuses on the control of microgrids where both gas and electricity are provided to the final customer, i.e., multi-carrier microgrids. Hence, these microgrids include thermal and electrical loads, renewable energy sources, energy storage systems, heat pumps, and combined heat and power units. The parameters characterizing the multi-carrier microgrid are subject to several disturbances, such as fluctuations in the provision of renewable energy, variability in the electrical and thermal demand, and uncertainties in the electricity and gas pricing. With the aim of accounting for the data uncertainties in the microgrid, we propose a Robust Model Predictive Control (RMPC) approach whose goal is to minimize the total economical cost, while satisfying comfort and energy requests of the final users. In the related literature various RMPC approaches have been proposed, focusing either on electrical or on thermal microgrids. Only a few contributions have addressed the robust control of multi-carrier microgrids. Consequently, we propose an innovative RMPC algorithm that employs on an uncertainty set-based method and that can provide better performance compared with deterministic model predictive controllers applied to multi-carrier microgrids. With the aim of mitigating the conservativeness of the approach, we define suitable robustness factors and we investigate the effects of such factors on the robustness of the solution against variations of the uncertain parameters. We show the effectiveness of the proposed RMPC approach by applying it to a realistic residential multi-carrier microgrid and comparing the obtained results with the ones of a baseline robust method. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This work is motivated by the emerging need for effective energy management approaches in multi-carrier microgrids. The inherent difficulty of scheduling simultaneously the operations of various energy infrastructures (e.g., electricity, natural gas) is exacerbated by the inevitable presence of uncertainties that affect the inter-dependent dynamics of different energy resources and equipment. The proposed robust MPC-based control strategy allows the energy manager to effectively determine an optimal energy scheduling of multi-faceted system components, making a tradeoff between performance and protection against data uncertainty. The presented strategy is comprehensive and generic, as it can be applied to different microgrid frameworks integrating various types of system components and sources of uncertainty, while at the same time being implementable in any energy management system.

Online optimal dispatch based on combined robust and stochastic model predictive control for a microgrid including EV charging station

To achieve carbon neutrality and meet the increased charging demand of electric vehicles, microgrids incorporating renewable energy and charging stations are considered one of the potential solutions. However, the inevitable uncertainties of renewable energy and load demand become a challenge for the online charging dispatch of the microgrid. Considering these uncertainties, this paper proposes a two-stage optimal framework for the online dispatch of a grid-connected DC microgrid. The first stage presents a power coordination model to obtain the schedule plans of the main grid, the energy storage unit and the charging station, where the combined robust and stochastic model predictive control approach with different granular models is developed to solve this problem and effectively deal with these uncertainties. In the second stage, the charging station allocation model is designed to determine the charging power for every EV, which takes into account the max-min fairness of the charging power. Numerical cases in the presence of uncertainties are studied to evaluate the proposed dispatch framework and the solving approaches. The simulation results show its superiority in both computational efficiency and operating cost.

Event‐triggered robust model predictive control for linear discrete‐time systems with a guaranteed average inter‐execution time

AbstractAn event‐triggered robust model predictive control (MPC) approach is proposed for linear discrete‐time systems with bounded disturbances. According to the probability distribution of bounded disturbances, an event‐triggered scheme involving a designed minimal robust positively invariant set is constructed to generate dynamic triggering sets. The MPC‐related optimization problem subject to hard constraints should be solved only at event‐triggered instants when the state is outside the corresponding triggering set. A classical tube‐based MPC that allows the initially predicted state different from the current actual state of the plant is considered to improve the feasible region. The designed event‐triggered controller can achieve a prescribed expectation of inter‐execution times and reduce the burden of communication and computation, while not sacrificing the quadratic performance significantly. It is proved that the proposed control approach ensures recursive feasibility and robust stability. Three examples are used to demonstrate the effectiveness and advantages of the proposed method.

Robust Resource-Aware Self-Triggered Model Predictive Control

The wide adoption of wireless devices in the Internet of Things requires controllers that are able to operate with limited resources, such as battery life. Operating these devices robustly in an uncertain environment, while managing available resources, increases the difficultly of controller design. This paper proposes a robust self-triggered model predictive control approach to optimize a control objective while managing resource consumption. In particular, a novel zero-order-hold aperiodic discrete-time feedback control law is developed to ensure robust constraint satisfaction for continuous-time linear systems.

Hybrid Physics-based and Data-driven Model Predictive Control for Multi-Zone Building's Thermal Comfort Under Disjunctive Uncertainty

This paper develops a hybrid approach that utilizes the distributional information of the disjunctive uncertainty sets and incorporates them into the model predictive control (MPC). This approach aims at the multi-zone building control to the thermal comfort, and it's robust to the uncertain weather forecast errors. The control objective is to maintain each zone's temperature and relative humidity within the specified ranges using the minimum cost of energy of the underlying heating system. The hybrid model is constructed using a physics-based and regression method for the temperature and relative humidity of each zone in the building. The uncertainty space is based on historical weather forecast error data, which are captured by a group of disjunctive uncertainty sets using k-means clustering algorithm. Machine learning approaches based on principal component analysis and kernel density estimation are used to construct each basic uncertainty set and reduce the conservatism of resulting robust control action under disturbances. A robust MPC framework is developed based on the proposed hybrid model and data-driven disjunctive uncertainty set. An affine disturbance feedback rule is employed to obtain a tractable approximation of the robust MPC problem. A case study of controlling temperature and relative humidity of a multi-zone building in Ithaca, New York, USA, is presented. The results demonstrate that the proposed hybrid approach can reduce 9.8% to 17.9% of total energy consumption compared to conventional robust MPC approaches. Moreover, the proposed hybrid approach can essentially satisfy the thermal constraints that certainty equivalent MPC and robust MPC largely violate.

Robust MPC for Systems With Model Uncertainties and Measurement Outliers

This paper considers the observer-based output feedback robust model predictive control (RMPC) problem for systems with model uncertainties and possible measurement outliers. For the sake of alleviating the effects from possible abnormal measurements, we design a set of observer-based output feedback RMPC controllers with the saturation constraint where the saturation level is adaptive according to the estimation errors. The purpose of the addressed problem is to design a set of desired RMPC controllers so as to guarantee the robustness and the asymptotical stability of the closed-loop system. Sufficient stability conditions are obtained by solving a time-varying terminal constraint set of an auxiliary optimization problem, and the corresponding control law and the upper bound of the quadratic cost function are derived. In addition, an algorithm including both off-line and on-line parts is provided to find a sub-optimal solution. Finally, two simulation examples are employed to illustrate the effectiveness of the proposed RMPC approach.