Min-max Model Predictive Control Research Articles

Lyapunov-based robust model predictive tracking controller design for discrete-time linear systems with ellipsoidal terminal constraint

In this article, a robust model predictive control (MPC) method is introduced based on Lyapunov-theory to control the discrete-time uncertain linear systems due to external disturbances. The structure of the proposed controller consists of two parts designed based on the offline/online schemes. In the offline-scheme, an H ∞ -based robust tracking controller is provided to satisfy the robust and tracking performance. Based on the H ∞ performance, a new linear matrix inequality problem is developed to calculate the offline-controller gains. By considering the Lyapunov-function of the offline-controller as the terminal cost of the MPC and the designed offline-controller, the online optimisation problem (OOP) is structured. This means, the MPC is designed based on the stabilised system to satisfy the physical limitations of the overall controller and the system states. Moreover, to improve the feasibility problem of the OOP, an ellipsoidal terminal constraint is added to the proposed MPC problem. Therefore, compared with the existing min–max MPC and tube MPC, the advantages of the overall controller are feasibility improvement and reducing the computational complexity with less conservatism while the robustness against parametric uncertainties and disturbances is achieved. Two simulation examples are employed to show the superiorities of the proposed robust MPC.

Two-dimensional optimization design of constrained minmax model predictive tolerant-fault control for nonlinear batch processes

The presence of actuator faults and disturbances, coupled with the nonlinear characteristics inherent in batch processes, poses challenges to the design of high-precision control algorithms, especially for variables that vary over a large range. To solve this issue, we propose a two-step 2D constrained model predictive fault-tolerant control method that integrates the theory of min-max optimization. First, the original nonlinear batch process is approximated at different operating points and transformed into multiple polytopic systems (MPSs), which includes the “worst case” scenario, i.e., a case with all internal effects. Next, according to its characteristics and by introducing an output tracking error and batch state differences, an equivalent model of a MPS is created and an iterative learning control law was used. Based on this model, a quadratic performance index function, including anti-external disturbances, is designed, which is divided into two parts covering the entire infinite horizon. The linear state feedback control law plus the initial free control moves comprise the control law. In this way, the optimal control inputs are generated, and the system can be controlled with a greater margin, improving feasibility. Moreover, for closed-loop stability, a parameter-dependent Lyapunov function is constructed. An allowable range of actuator faults that the designed controller can resist is also considered and constructed. The advantage of the proposed control algorithm is that it fully considers the repetitive characteristics of batch processes, and for the first time, a 2D two-step control strategy related to time and batch directional information is designed, which greatly improves the system output tracking performance. Using a nonlinear batch reactor as an example, the method proposed in this study demonstrates its superiority tracking performance and ability to resist external disturbances than that of the traditional method.

Handling the Constraints in Min-Max MPC

One of the major sources of conservativeness in min-max model predictive control (MPC) is the handling of constraints, where the ellipsoidal robust invariant set is utilized for the proposal of sufficient and conservative conditions for the satisfaction of the constraints. In this article, in order to reduce the conservativeness due to the constraint handling, we add additional relaxation variables to the physical constraints and propose a two-step approach through relaxing the constraints by the amount which is determined by the calculating of the maximal admissible set (MAS). The constraints are relaxed in an iterative manner to avoid the constraints violation, and the constraint relaxation variables are degrees of freedom for relaxing the constraints and improving the control performance. Moreover, we show that under certain circumstance, the physical constraints can be removed without the constraint violation. The proposed approach is shown to be recursively feasibility and its effectiveness is verified through an air conditioning control in a building energy system. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —Buildings are account for large percentage of worldwide energy consumptions. One of the most applicable method for optimization of building energy system subject to multiple constraints is the model predictive control (MPC). However, the industrial MPC is usually not recursively feasible, which implies that the optimization problem can become infeasible and the software will be terminated at some time. In order to apply the MPC synthesis approach (MPC with recursive feasibility guarantee), we have to overcome the conservativeness problem due to the handling of constraints. We propose a useful approach in this work by introducing relaxation variables which act as degrees of freedom for improving the control performance, while the physical constraints are still satisfied. The proposed approach is verified through an example of a 24m <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$^2$</tex-math> </inline-formula> office room located in Cyber-Physical Energy System (CPES) lab in Western China Science and Technology Innovation Harbour in Xianyang, China. The numerical results show the performance improve of the proposed approach.

Efficient min–max MPC: Achieving a large domain of attraction with short horizon

AbstractThis article considers the regulation problem of a constrained linear system with bounded disturbance. The objective is to overcome the offline and online problems of the standard dual‐mode min–max model predictive control (MPC). The offline problem is related to the construction of a polyhedral terminal set, which is prohibitively complex especially for high order systems. We propose to replace the polyhedral set with the ellipsoidal one, which is much easier to construct. The online problem is related to the computational burden, which grows exponentially with the prediction horizon . We show how to employ a new terminal cost function, that results in a new min–max MPC with a large domain of attraction even with or . Hence the computational complexity is drastically reduced. Two numerical examples with comparison to earlier solutions from the literature illustrate the effectiveness of the proposed approach.

Robust Precision Position Tracking of Planar Motors Using Min-Max Model Predictive Control

In this article, a min-max model predictive control (MPC) method of planar motors is proposed for the first time to achieve robust precision position tracking, which has a low computational burden and strong capability to deal with the problems of stability, robustness, optimization, and input constraints. A state-space model with a homogeneous state equation is built to describe the dynamics of the time-varying reference trajectory. Combining the state-space model of the reference trajectory and that of the planar motor, an augmented state-space model is established to obtain an error state formulation. Then, using the error state formulation, a min-max optimal control problem subject to the constraints on bounded uncertainty, stability, and control input is developed. Moreover, applying the theory of asymptotically stable invariant ellipsoids and employing the nested invariant ellipsoids, the explicitly linear state-feedback control laws are obtained using a linear-matrix-inequalities based offline control algorithm. Finally, the min-max MPC is applied to a planar motor system developed in the laboratory for an experimentally comparative study. The results demonstrate the effectiveness of the proposed min-max MPC of planar motor for robust precision position tracking applications.

Distributed Model Predictive Control for Vehicle Platoon With Mixed Disturbances and Model Uncertainties

To successfully implement the platoon control of connected and automated vehicles (CAVs), the model uncertainties, external disturbances, and time delays must be addressed. In this study, we propose a distributed Model Predictive Control scheme to achieve offset-free tracking for CAVs with model uncertainties and mixed disturbances which contains both stochastic and deterministic noises. A simple Vehicle-to-Vehicle communication flow with the short-distance and low-volume information exchange is adopted to reduce the effect of the communication delay and dropout. First, the deterministic and stochastic perturbances are addressed separately in constraints handling with an integrated control law based on the mismatched prediction model. Additional constraints on coupling input and output are imposed to guarantee the assumption about forecast trajectory. Then an observer gives an unbiased estimate of the state and integrated disturbance. Based on the estimation, the target calculator eliminates the steady tracking offset by designing a new target in real-time and the Min-Max Model Predictive Control steers the uncertain systems to the new target under the worst case within the domain of disturbances. Lastly, the Optimal Control Problem is reformulated and can be efficiently solved by existing solvers. Simulation results indicate that the proposed method shows superiority in interference rejection and offset-free tracking.

Robust Min-Max Model Predictive Vehicle Platooning With Causal Disturbance Feedback

Platoon-based vehicular cyber-physical systems have gained increasing attention due to their potentials in improving traffic efficiency, capacity, and saving energy. However, external uncertain disturbances arising from mismatched model errors, sensor noises, communication delays and unknown environments can impose a great challenge on the constrained control of vehicle platooning. In this paper, we propose a closed-loop min-max model predictive control (MPC) with causal disturbance feedback for vehicle platooning. Specifically, we first develop a compact form of a centralized vehicle platooning model subject to external disturbances, which also incorporates the lower-level vehicle dynamics. We then formulate the uncertain optimal control of the vehicle platoon as a worst-case constrained optimization problem and derive its robust counterpart by semidefinite relaxation. Thus, we design a causal disturbance feedback structure with the robust counterpart, which leads to a closed-loop min-max MPC platoon control solution. Even though the min-max MPC follows a centralized paradigm, its robust counterpart can keep the convexity and enable the efficient and practical implementation of current convex optimization techniques. We also derive a linear matrix inequality (LMI) condition for guaranteeing the recursive feasibility and input-to-state practical stability (ISpS) of the platoon system. Finally, simulation results are provided to verify the effectiveness and advantage of the proposed MPC in terms of constraint satisfaction, platoon stability and robustness against different external disturbances.

Robust Multi-Model Predictive Control via Integral Sliding Modes

This paper presents a novel optimal control approach for systems represented by a multi-model, i.e., a finite set of models, each one corresponding to a different operating point. The proposed control scheme is based on the combined use of model predictive control (MPC) and first order integral sliding mode control. The sliding mode control component plays the important role of rejecting matched uncertainty terms possibly affecting the plant, thus making the controlled equivalent system behave as the nominal multi-model. A min-max multi-model MPC problem is solved using the equivalent system without further robustness oriented add-ons. In addition, the MPC design is performed so as to keep the computational complexity limited, thus facilitating the practical applicability of the proposal. Simulation results show the effectiveness of the proposed control approach.

An adaptive model predictive control strategy for a class of discrete‐time linear systems with parametric uncertainty

SummaryIn this study, an adaptive model predictive control (MPC) strategy is proposed for a class of discrete‐time linear systems with parametric uncertainty. In the presented adaptive MPC, an updating law is firstly designed to update the estimated parameters online. By utilizing the estimated parameters, a standard MPC optimization problem for unconstrained systems is established. Then, to deal with constrained systems, the min–max MPC technique is developed under the set‐based approach for the estimated parameters. Furthermore, it is shown theoretically that the recursive feasibility and closed‐loop stability can be rigorously proved, respectively. Finally, numerical simulations and comparative analysis are presented to illustrate the superiority of the proposed algorithms in control performance.

Reducing the computational effort of min-max model predictive control with regional feedback laws

Abstract Recently, a regional MPC approach has been proposed that exploits the piecewise affine structure of the optimal solution (without computing the entire explicit solution before). Here, regional refers to the idea of using the affine feedback law that is optimal in a vicinity of the current state of operation, and therefore provides the optimal input signal without requiring to solve a QP. In the present paper, we apply the idea of regional MPC to min-max MPC problems. We show that the new robust approach can significantly reduce the number of QPs to be solved within min-max MPC resulting in a reduced overall computational effort. Moreover, we compare the performance of the new approach to an existing robust regional MPC approach using a numerical example with varying horizon. Finally, we provide a rule for choosing a suitable robust regional MPC approach based on the horizon.

Robust event‐triggered distributed min–max model predictive control of continuous‐time non‐linear systems

Due to the features of event-triggered control in exploiting and saving system resources, they have been widely applied in sensor networks, multi-agent systems, networked control systems and so on. In this study, the authors focused on robust event-triggered distributed model predictive control (RETDMPC). Subject to disturbances and parametric uncertainties, they first applied the min–max model to RETDMPC. The min–max RETDMPC methodology is used to guarantee the robustness of the system state by taking the worst possible case of unknown uncertainties into consideration. Furthermore, in this framework, a new cost function is developed in which unknown uncertainties are considered. Next, sufficient conditions are provided to ensure the feasibility and stability of their developed min–max RETDMPC. Finally, a practical example is given to illustrate the advantages of their algorithm by comparing to the conventional model predictive control.

Min-Max Model Predictive Vehicle Platooning With Communication Delay

Vehicle platooning gains its popularity in improving traffic capacity, safety and fuel saving. The key requirements of an effective platooning strategy include keeping a safe inter-vehicle space, ensuring string stability and satisfying vehicular constraints. To meet these requirements, this paper proposes a distributed min-max model predictive control (MPC). One technical contribution is that the proposed MPC can guarantee input-to-state predecessor-follower string stability, in the presence of vehicle-to-vehicle communication delays and realistic constraints. Another technical contribution is the development of a new concept of input-to-state stability margin for analyzing the platooning system that is nonlinear under MPC. The proposed MPC is applicable to both homogeneous and heterogeneous platoons because only the point-mass vehicle model is needed. The proposed MPC also has reduced communication burden because each vehicle in the platoon only transmits its current acceleration to the adjacent follower. The design efficacy is verified by simulating a platoon composed of five vehicles under different uncertainties and communication delays.

On tracking control problem for polysolenoid motor model predictive approach

The Polysolenoid Linear Motor (PLM) have been playing a crucial role in many industrial aspects due to its functions, in which a straight motion is provided directly without mediate mechanical actuators. Recently, with several commons on mathematic model, some control methods for PLM based on Rotational Motor have been applied, but position, velocity and current constraints which are important in real systems have been ignored. In this paper, position tracking control problem for PLM was considered under state-independent disturbances via min-max model predictive control. The proposed controller forces tracking position errors converge to small region of origin and satisfies state including position, velocity and currents constraints. Further, a numerical simulation was implemented to validate the performance of the proposed controller.

Constrained Model Predictive Control for Nonlinear Markov Jump System With Persistent Disturbance via Quadratic Boundedness

In this article, a robust quadratic-boundedness-based model predictive control (MPC) scheme, for a discrete-time nonlinear Markov jump system (MJS), is extended to the case with persistent bounded disturbance and nonhomogeneous transition probability. By applying the S-procedure, the constraint conditions, the persistent bounded disturbance and the sufficient stability conditions are all derived in term of a few linear matrix inequalities (LMIs), thus the original min-max optimization problem is transformed into a convex optimization problem in LMI paradigm. At each sampling time, the control moves satisfying the control constraint are obtained online and implemented in the nonlinear MJS. Quadratically boundedness and min-max MPC are combined to achieve the closed-loop stochastic stability of the controller with respect to the persistent bounded disturbance. A numerical example is presented to demonstrate the effectiveness of the proposed results.

Formulation of min-max model predictive control as a box-constrained robust least squares estimation problem

The purpose of this paper is to propose a new approach to the Min-Max Model Predictive Control (MMMPC) of Linear Time-Invariant Discrete-time Polytopic (LTIDP) systems. The purpose is to simplify the treatment of complex issues like stability and feasibility analysis of robust MPC as well as to reduce the complexity of the relative optimization procedure. The new approach is based on a two Degrees Of Freedom (2DOF) control scheme where the output r(k) of the feedforward Input Estimator (IE) is used as input forcing a stable closed-loop system Σf. Σf is the feedback connection of an LTIDP plant Σp with an LTI dynamic controller Σg. The task of Σg is to guarantee the quadratic stability of Σf, as well as the fulfillment of hard constraints on some physical variables of Σf for any input r(k) satisfying an ”a priori” determined admissibility condition. The input r(k) is computed by the feedforward IE through the on-line minimization of a worst case finite-horizon quadratic cost functional and is applied to Σf according to the usual receding horizon strategy. Rather than resorting to an ”ad hoc” software, the numerical complexity issue is here addressed reducing the number of both decision variables and constraints involved in the on-line constrained optimization procedure. This is obtained modeling r(k) as a B-spline function, which is known to be a universal approximator which also admits a parsimonious parametric representation. This allows us to reformulate the minimization of the worst case cost functional as a box-constrained Robust Least Squares (RLS) estimation problem which can be efficiently solved using Second Order Cone Programming (SOCP).

Distributed min–max MPC for dynamically coupled nonlinear systems: A self-triggered approach

This paper considers the formation stabilization problem of dynamically coupled nonlinear systems with parametric uncertainties and additive disturbances. We develop a distributed min–max model predictive control (MPC) framework, in which each subsystem adopts the local optimal control action by solving the constrained optimization problem. As a main contribution of this paper, a self-triggered strategy is presented within the proposed framework for resource-constrained coupled systems. By implementing the distributed self-triggered scheduler, the communication burden is significantly alleviated while ensuring comparable control performance. In addition, for each subsystem the latest information is transmitted to its neighbors asynchronously at the local triggering time instants. Moreover, the resulting distributed self-triggered min–max MPC framework ensures the constraints satisfaction and the closed-loop stability. Finally, the numerical experiments are performed to verify the theoretical results.

Adaptive MPC for constrained systems with parameter uncertainty and additive disturbance

In this study, the authors propose an adaptive model predictive control (MPC) algorithm for constrained linear systems in state space subject to uncertain model parameters and disturbances. An iterative set membership identification algorithm is first presented to update the uncertain parameter set at each time step. Based on the shrunken uncertain parameter set, an MPC controller is then designed to robustly stabilise the uncertain systems subject to state and input constraints. The algorithm can efficiently reduce the size of the uncertain parameter set in min-max MPC setting, and therefore improve the control performance. The algorithm is proved to ensure constraint satisfaction, recursive feasibility and input-to-state practical stability of the closed-loop system even in the presence of system uncertainties. A numerical example and a brief comparison with traditional min-max MPC are provided to demonstrate the efficiency of the proposed algorithm.

Min Max Model Predictive Control for Polysolenoid Linear Motor

&lt;p&gt;&lt;span lang="EN-US"&gt;The Polysolenoid Linear Motor (PLM) have been playing a crucial role in many industrial aspects because it provides a straight motion directly without mediate mechanical actuators. Some control methods for PLM based on Rotational Motor are applied to obtain several good performances, but position and velocity constraints which are important in real systems are ignored. In this paper, we analysis control problem of tracking position in PLM under state-independent disturbances via min-max model predictive control. The proposed controller brings tracking position error converge to zero and satisfies state including position and velocity and input constraints. The simulation results validity a good efficiency of the proposed controller.&lt;/span&gt;&lt;/p&gt;

An efficient offline implementation for output feedback min‐max MPC

SummaryPrevious works have presented the output feedback min‐max model predictive control (MPC) for the discrete‐time system with both polytopic uncertainty and bounded persistent disturbance, where the controller parameters are optimized at each sampling instant. This paper proposes the corresponding offline approach in order to reduce the online computational burden. Such offline MPC, when the state is measurable and there is no disturbance, has been constructed in the work of Wan and Kothare (An efficient off‐line formulation of robust model predictive control using linear matrix inequalities. Automatica. 2003;39(5):837‐846). Since this paper considers the case when the true state is unknown, the ellipsoidal regions of attraction (applying only to the estimated state) lose their asymptotic invariance property, and the estimation error set (EES) has a major effect on the control performance. This paper refreshes EES invoking the one‐step reachable set and guarantees that the signals being penalized in the performance cost function to converge to a neighborhood of the equilibrium point. Two examples are given to illustrate the effectiveness of the approach.

Robust self-triggered min–max model predictive control for discrete-time nonlinear systems

In this paper, we propose a robust self-triggered model predictive control (MPC) algorithm for constrained discrete-time nonlinear systems subject to parametric uncertainties and disturbances. To fulfill robust constraint satisfaction, we take advantage of the min–max MPC framework to consider the worst case of all possible uncertainty realizations. In this framework, a novel cost function is designed based on which a self-triggered strategy is introduced via optimization. The conditions on ensuring algorithm feasibility and closed-loop stability are developed. In particular, we show that the closed-loop system is input-to-state practical stable (ISpS) in the attraction region at triggering time instants. In addition, we show that the main feasibility and stability conditions reduce to a linear matrix inequality for linear case. Finally, numerical simulations and comparison studies are performed to verify the proposed control strategy.