Robust Model Predictive Control Algorithm Research Articles

An improved gain-scheduling robust MPC for path following of autonomous independent-drive electric vehicles with time-varying and uncertainties

This paper proposes a gain-scheduling robust model predictive control (GS-RMPC) algorithm for the path-following problem of autonomous independent-drive electric vehicles (AIDEVs) with consideration of time-varying and uncertainties. Firstly, the polytopic uncertainty method and norm-bounded uncertainty method are introduced to characterise the vehicle dynamics model. Secondly, the infinite predict horizon optimisation process of online GS-RMPC is transformed into a series of linear matrix inequalities (LMIs) by minimising the worst-case objective function while considering all scheduling states in the polytope. Thirdly, an offline solution is also proposed to reduce the computational burden based on asymptotically stable invariant ellipse sets. Then, a hierarchical control structure is proposed to distribute the additional yaw moment, and a multi-step predictor is designed to compensate for the actuator time delay. Finally, the hardware-in-the-loop (HIL) testing is conducted to verify the efficacy of the proposed strategy.

Convex distributed robust model predictive control for collision and obstacle avoidance

AbstractThe article proposes a convex distributed robust model predictive control algorithm for collision avoidance in multi‐agent systems with additive disturbances, which utilizes the concept of tube model predictive control to address the disturbances. To tackle the coupled collision avoidance constraints, each agent integrates assumed nominal position and input trajectories from its neighbors, rather than relying on actual ones. Compatibility constraints are then designed using the normal vector of the constructed separating hyperplanes for collision avoidance and the residual collision avoidance margin of the optimal solutions at the last time step to restrict deviations between assumed and actual trajectories and ensure consistency among the agents. The nonconvex collision and obstacle avoidance constraints are convexified using the concept of safe sets, transforming them into time‐varying closed polyhedral constraints while accounting for the impact of disturbances. Particularly, the residual collision avoidance margin is incorporated into the construction of safe sets for the satisfaction of coupled collision avoidance constraints. Consequently, the original collision avoidance optimal control problems can be efficiently and simultaneously solved for all agents using standard quadratic programming techniques. Under the design of local terminal constraint sets for each agent, the proposed algorithm ensures a rigorous analysis of robust constraint satisfaction for collision avoidance constraints, as well as an assessment of recursive feasibility and closed‐loop stability. The effectiveness of the algorithm is demonstrated through an illustrative example, and simulation results validate its ability to successfully achieve collision and obstacle avoidance even in the presence of disturbances.

Robust model predictive control for perturbed nonlinear systems via an error differential-integral based event-triggered approach

This paper aims to establish a new event-triggered model predictive control (MPC) strategy for perturbed nonlinear systems working in a networked environment, which can effectively save the computation and communication resources while ensuring control performance. The core of the framework is a new event-triggered mechanism, which is built by combining the differential and integral (D-I) information of the error between the optimal state and the actual state in a pre-specified time horizon. Based on such a triggering mechanism, a D-I based event-triggered robust MPC algorithm is proposed to stabilise the system and reduce the computation/communication resource consumption. In addition, sufficient conditions to ensure the feasibility and stability of the proposed MPC algorithm and avoid Zeno behaviour are obtained through rigorous theoretical analysis. Finally, the proposed control framework is implemented in the nonlinear cart-damper-spring and autonomous underwater vehicle systems to verify its effectiveness.

A Distributionally Robust Optimization Approach to Two-Sided Chance-Constrained Stochastic Model Predictive Control With Unknown Noise Distribution

In this work, we propose a distributionally robust stochastic model predictive control (DR-SMPC) algorithm to address the problem of multiple two-sided chance constrained discrete-time linear systems corrupted by additive noise. The prevalent mechanism to cope with two-sided chance constraints is the so-called risk allocation approach, which conservatively approximates the two-sided chance constraints with two single chance constraints by applying Bool's inequality. In this proposed DR-SMPC framework, an exact second-order cone approach is adopted to abstract the multiple two-sided chance constraints by considering the first and second moments of the noise. With the proposed DR-SMPC algorithm, the worst-case probability of violating safety constraints is guaranteed to be within a pre-specified maximum value. By flexibly adjusting this pre-specified maximum probability, the feasible region of the initial state can be increased for the SMPC problem. The recursive feasibility and convergence of the proposed DR-SMPC are rigorously established by introducing a binary initialization strategy for the nominal state. A simulation study of a single spring and double mass system was conducted to demonstrate the effectiveness of the proposed DR-SMPC algorithm.

SD-ARX modeling and robust MPC with variable feedback gain for nonlinear systems

As a generalized input-output model, the state-dependent exogenous variable autoregressive (SD-ARX) model has been intensively utilized to model complex nonlinear systems. Considering that more freedom can be provided by the state feedback control with variable feedback gain for constructing robust controllers, we propose a robust model predictive control (RMPC) algorithm with variable feedback gain on the basis of the SD-ARX model. First, the polytopic state space models (SSMs) of the system are constructed and the prediction accuracy of the SSMs is further improved by using the parameter variation rate information of the SD-ARX model. Then, an RMPC algorithm with variable feedback gain is synthesized for increasing the design freedom and enhancing the control performance. Two simulation examples, that is, the modeling and control of a continuous stirred tank reactor (CSTR) and a water tank system, are provided to demonstrate the feasibility and effectiveness of the proposed RMPC algorithm.

Robust tube-based NMPC for dynamic systems with discrete degrees of freedom

We present a robust nonlinear model predictive control (NMPC) algorithm for dynamic systems with mixed degrees of freedom, i.e., continuous and discrete manipulated variables. The discrete variables are often excluded from the NMPC layer and determined at a supervisory layer, which typically leads to suboptimal solutions. In contrast, the proposed controller optimizes both classes of degrees of freedom within the NMPC layer, thereby enhancing the closed-loop performance. Our approach relies on a computationally efficient relaxation and integrality restoration strategy. Initially, a nominal controller computes relaxed central reference trajectories for the states and the input variables. These trajectories are then tracked by an ancillary NMPC controller before the integrality of the discrete variables is restored. The role of the ancillary controller is to ensure robustness against the integer restoration error as well as other disturbances based on a nonlinear Tube MPC approach. Furthermore, we provide sufficient conditions to establish recursive feasibility and guarantee robust closed-loop stability. Finally, the effectiveness of the approach is demonstrated through two nonlinear simulation examples, affirming its viability and robustness.

Self-triggered robust output feedback model predictive control of discrete-time linear systems

In networked control systems, the communication load is a main concern for implementing model predictive control (MPC). This paper introduces a self-triggered MPC algorithm under network environments to reduce communication load, for discrete-time linear systems with bounded state and output disturbances, when only the system output can be measured at triggering instants. The proposed algorithm mainly based on a state estimator whose estimation error is bounded by an invariant set, and on the self-triggered model predictive control of a nominal system. Moreover, the new cost function explicitly adopts communication load is in consideration. The proposed algorithm is proved to drive the system to an invariant set with respect to the bounded disturbances. Finally, simulation results is provided to verify the effectiveness of the proposed robust output feedback MPC algorithm.

Dual-Mode Robust Fuzzy Model Predictive Control of Time-Varying Delayed Uncertain Nonlinear Systems With Perturbations

For time-varying delayed nonlinear systems with parameter uncertainties and persistent disturbances, an online and an offline robust fuzzy model predictive control (RFMPC) algorithms are proposed in this paper. Both methods guarantee the input-to-state stability (ISS) of the system. Furthermore, a novel alternative optimization (AOP) approach and a dual-mode OP/AOP strategy are proposed to prevent the performance deterioration of the online optimal control (OP) approach due to the challenges in addressing the bilinear matrix inequality (BMI) constraints. With the established AOP and dual-mode OP/AOP techniques, the optimization problem constrained by BMIs is transformed into convex, and a significantly more precise approximation of the ellipsoidal minimal robust positively invariant (EmRPI) set can be calculated. Besides, the system can eventually converge into a more compact ellipsoidal set. These two alternative optimization methods can be easily extended to various nonlinear systems. A numerical example and a continuous-time stirring tank (CSTR) example are provided to validate the effectiveness and advantages of the established methodologies.

A robust model predictive control algorithm for input–output LPV systems using parameter extrapolation

In this paper, we introduce a novel robust Model Predictive Control (MPC) algorithm for Linear Parameter Varying (LPV) systems represented in the Input–Output (IO) form. The proposed scheme embeds integral action and ensures output reference tracking for piece-wise constant signals. The algorithm is based on the online extrapolation of the LPV scheduling parameters, which are generated recursively based on a simple Taylor expansion argument. Closed-loop asymptotic stability, recursive feasibility of the online optimisation, as well as robustness towards bounded disturbances and scheduling parameter prediction uncertainties are demonstrated. Two distinct nonlinear multi-input multi-output benchmarks are used to illustrate the effectiveness of the proposed method: a numeric simulation example, used to compare the method to state-of-the-art techniques, and a high-fidelity twin rotor system, for which the method is further validated. Real-time capabilities of the proposed scheme are highlighted, since it only requires one Quadratic Program to be evaluated per discrete-time sample, during the implementation.

Robust predictive control of perturbed switched linear systems with stable and unstable subsystems in the presence of uncertainties

In this article, robust model predictive control is proposed for discrete-time switched linear systems in the presence of external disturbances, model uncertainties and constraints on the states and input. To solve the problem, a mode-dependent average dwell time switching strategy is constructed in controller design procedure, with less limitations than the existing average dwell time switching. Using some mathematical tools, the model predictive control problem is turned into a feasible linear matrix inequality. The control algorithm is presented via two theorems, and the stability analysis is analytically presented, based on multiple Lyapunov function method. Compared with the previous studies, the underlying switched system may consist of stable and unstable subsystems, perturbed by model uncertainties and external disturbances. The simulation results are also given to study the effectiveness of the proposed robust model predictive control algorithm, from a comparison viewpoint.

MPC-Based Cooperative Enclosing for Nonholonomic Mobile Agents Under Input Constraint and Unknown Disturbance.

In this article, a model predictive control (MPC)-based cooperative target enclosing control approach is investigated for multiple nonholonomic mobile agents with input constraints and unknown disturbances. The agents are required to move along a desired circular orbit centered at a stationary target and maintain an even distribution on the orbit. Based on a dual-mode MPC strategy, a cooperative target enclosing control law is designed by only using the local sensing information. When the agents are inside a terminal region, a locally cooperative stabilizing control law is designed with a signal function defined componentwise part compensating for the unknown disturbances. A robust MPC algorithm is designed for the agents to enter the terminal region in finite time. Global asymptotic stability is guaranteed for multiple nonholonomic mobile agents with input constraints and unknown disturbances. Simulation results illustrate the effectiveness of the proposed approach.

Motion control framework for unmanned wheel-legged hybrid vehicle considering uncertain disturbances based robust model predictive control

The paper proposes a motion control framework for the unmanned wheel-legged hybrid vehicle to track the motion trajectory considering uncertain disturbances. The whole-body dynamic model is built with the contact force of each rolling wheel, which serves as the foundation to obtain trajectory tracking. The angular momentum and linear momentum are optimized by the robust model predictive control algorithm considering the soft constraint of the relaxation variable. The contact force between wheel and ground is solved by the quadratic programming algorithm to efficiently obtain the flexion/extension joint and wheel motion planning. Then, the explicit algorithm to calculate the torque commands of the flexion/extension joint considering the feed-forward torque and feedback torque to improve the control accuracy. Simulation results validate that the control framework based on the robust model predictive control algorithm can solve the uncertain disturbances in process of the vehicle running on the rough road.

A Distributionally Robust Optimization Based Method for Stochastic Model Predictive Control

Two stochastic model predictive control algorithms, which are referred to as distributionally robust model predictive control algorithms, are proposed in this article for a class of discrete linear systems with unbounded noise. Participially, chance constraints are imposed on both of the state and the control, which makes the problem more challenging. Inspired by the ideas from distributionally robust optimization (DRO), two deterministic convex reformulations are proposed for tackling the chance constraints. Rigorous computational complexity analysis is carried out to compare the two proposed algorithms with the existing methods. Recursive feasibility and convergence are proven. Simulation results are provided to show the effectiveness of the proposed algorithms.

An algorithm for the selection of linearisation points in non-linear systems: a diesel air-path case study

One typical approach to implement model predictive control in relatively fast sample rate applications utilises linearised plant models to simplify the online optimisation problem in the presence of limited computation budget. A number of linearisation points may be used to account for the variation in the non-linear plant over the entire operating domain, leading to potential switched linear time-invariant (LTI) or linear time-variant (LTV) implementations. The choice of these linearisation points, and the regions over which they are considered valid, can have a significant impact on the closed loop performance of the overall controller. In this work, in the presence of a high-fidelity digital twin of a non-linear plant, a numerical algorithm for the selection of linearisation points is proposed. The approach makes use of the concept of disturbance sets to determine both the linearisation points and the domain to which each linearisation point applies. The use of disturbance sets provides an implicit connection to closed-loop controller performance if robust model predictive control algorithms are implemented. The proposed approach is implemented for the challenging problem of diesel air-path control, with simulation results on a high-fidelity mean value engine model used to demonstrate the efficacy of the approach.

Cloud-based model-predictive-control of a battery storage system at a commercial site

Reducing costs for battery energy storage systems and the increasing availability of onsite generation sources are driving development of complex battery control algorithms principally aimed at minimising electricity costs. These algorithms combine forecasts of site consumption, generation and electricity costs and a model of the battery system in a solver that minimises a cost objective over some forecast horizon. Often they are trialled in simulation environments without the complexities introduced by real-world deployments such data quality and reliability issues, communication issues and forecast inaccuracies to name a few. This study reports on a trial demonstration of a cloud-based data-driven robust model predictive battery control algorithm controlling an existing 150kWh lithium-ion battery at an operational site housing 100 + office staff. Forecasting model and control performance are evaluated in situ. Despite two of the four battery inverters being non-functional, the algorithm delivered electricity bill cost savings of 5.5 %, of which two-thirds was the result of reducing the sites capacity charge from 358 kVa to 317 kVa. Throughout the trial multiple operational issues were encountered mostly related to data outages, equipment and communications reliability and the lessons learned from managing these occurrences are also discussed.

Attitude tracking control for reentry vehicles using centralised robust model predictive control

In this work, a centralised robust model predictive control (CRMPC) algorithm is proposed for reentry vehicles to track reference attitude trajectories subject to state/input constraints and uncertainties. In contrast to most designs that apply a cascade control structure for the two-timescale attitude dynamical systems, the proposed control scheme utilises a centralised structure to avoid additional controller development and parameter turning. By designing a nonlinear feedback law and tightening the system constraints, robust constraint satisfaction can be ensured for all admissible uncertainties. In addition, to guarantee the recursive feasibility and closed-loop stability of the proposed CRMPC, a terminal controller, along with a terminal region, is introduced. The validity of using the proposed approach to solve the considered problem is confirmed by executing several experimental studies, which were compared against two other established methods.

Dynamic Lane Tracking Control of the Commercial Vehicle Based on RMPC Algorithm Considering the State of Preceding Vehicle

In order to improve the adaptability of the lane keeping control system to complex environments, a dynamic lane tracking control strategy of the commercial vehicle based on the robust model predictive control (RMPC) algorithm is proposed considering the state of the preceding vehicle. An RMPC controller is designed with path deviation and control increment as the objective function. The model predictive control problem is transformed into a min–max optimization problem. The linear matrix inequality (LMI) is used for the optimal solution to obtain the optimal control quantity. The strategy to improve the safety and comfort dynamically in the process of lane keeping is designed by adjusting the weight coefficient matrix of RMPC based on fuzzy theory. The results of the simulation and HiL test show that the RMPC controller can meet the requirement of adjusting the lane tracking process dynamically according to the state of the preceding vehicle, which keeps the balance between safety and comfort.

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.

Nonlinear curve fitting-based fast robust MPC algorithm for nonlinear system

In the existing efficient robust model predictive control (ERMPC) algorithms (see e.g. [14,31,32]), through offline optimization and online lookup table calculation, a fixed state feedback control law or a linear interpolated control law is applied to a system when the system state lies between two adjacent polyhedrons, which undoubtedly will result in conservativeness of the controller. Faced with this issue, an improved ERMPC algorithm is proposed in this paper, which considers the nonlinearity between the state feedback control laws with respect to polyhedrons and the norm distance from system state to origin, and can provide continuously variable state feedback control law varying with the state. First, a set of polyhedral parameters and their corresponding state feedback control law sequences are obtained offline by solving a set of LMIs optimization problems. Next, for each state feedback control law sequence, a nonlinear fitting function is established offline between the state feedback control law and its serial number. Then a simplified lookup table is constructed offline to save memory space and shorten online computation time of the controller. According to the simplified lookup table and information of the norm distance from system state to origin, we online establish the coordinate of current state in the nonlinear fitting curve for getting current feedback control law, which changes continuously with the state. The proposed ERMPC algorithm is successfully applied to an actual fast-responding linear one stage inverted pendulum (LOSIP) system to verify its effectiveness.

Application Strategies of Model Predictive Control for the Design and Operations of Renewable Energy-Based Microgrid: A Survey

In recent times, Microgrids (MG) have emerged as solution approach to establishing resilient power systems. However, the integration of Renewable Energy Resources (RERs) comes with a high degree of uncertainties due to heavy dependency on weather conditions. Hence, improper modeling of these uncertainties can have adverse effects on the performance of the microgrid operations. Due to this effect, more advanced algorithms need to be explored to create stability in MGs’. The Model Predictive Control (MPC) technique has gained sound recognition due to its flexibility in executing controls and speed of processors. Thus, in this review paper, the superiority of MPC to several techniques used to model uncertainties is presented for both grid-connected and islanded system. It highlights the features, strengths and incompetencies of several modeling methods for MPCs and some of its variants regarding handling of uncertainties in MGs. This survey article will help researchers and model developers to come up with more robust model predictive control algorithms and techniques to cope with the changing nature of modern energy systems, especially with the increasing level of RERs penetration.