Model predictive control for uncertain max–min-plus-scaling systems

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.

On input-to-state stability of min–max nonlinear model predictive control

This work investigates the event‐driven model predictive control problem for perturbed nonlinear systems with input and state constraints. Firstly, a new event‐driven triggering condition, considering the gradient and accumulation of the error between the optimal and actual states at adjacent moments with a state‐dependent relative threshold is designed with a guaranteed positive inter‐event time to avoid Zeno behaviour. Secondly, an error gradient and accumulation type event‐driven model predictive control framework is designed based on the new triggering condition and the dual‐mode strategy, for effectively reducing the calculation and communication burden of the model predictive control controller while ensuring desired control performance. Furthermore, the feasibility and the input‐to‐state practical stability (ISpS) of gradient and accumulation‐type event‐driven model predictive control are rigorously guaranteed theoretically. At last, the simulation results are provided to validate the this algorithm.

MPC for continuous piecewise-affine systems

International audience

Torque distribution optimization of electric vehicles should consider various demands, including energy conservation, power, and tire antislip, which are usually coupled. Furthermore, there is a higher requirement for real-time feasibility of controllers. This study proposes an embedded model predictive controller called EmMPC and applies it to the torque distribution optimization of electric vehicles equipped with two or more motors. In EmMPC, barrier functions and normalized projected gradients are utilized to handle state inequality and linear input constraints. A fast line search method combined with heuristic search regions and quadratic function properties is presented to reduce computational burden further. Subsequently, an offline and online combined torque distribution optimization strategy is presented based on EmMPC and an analysis of maximum energy-saving potential. Finally, hardware-in-the-loop experiments were performed with an embedded vehicle-mounted controller to illustrate the effectiveness of our method.

SUMMARYThis paper addresses a new type of model predictive control problem for a hybrid system that consists of a continuous‐time linear system and a temporal/spatial directed graph, called a directed‐graph constrained system. Motivated by the obstacle avoidance problem, the problem is newly formulated, where the continuous‐time control input and the waypoints of the state are simultaneously optimized under a temporal/spatial directed graph as well as input/state linear constraints, and a method for efficiently solving this problem is developed. Numerical examples are presented to verify that the proposed approach is effective. Copyright © 2013 John Wiley &amp; Sons, Ltd.

Min-max Model Predictive Control of Nonlinear Systems: A Unifying Overview on Stability

The paper considers formulations of constrained model predictive control (MPC) problems for systems with dynamics defined on smooth manifolds. Generalizations of conventional nonlinear MPC stabilization techniques based on the terminal set and terminal penalty approaches are presented based on recent work by the authors. The significance of the results includes the possibility of systematically generating globally stabilizing, discontinuous, time-invariant feedback laws in situations when smooth, or even continuous, globally stabilizing, time-invariant stabilizers do not exist. To illustrate the development of nonlinear MPC on manifolds, the paper considers an example of a system with dynamics evolving on SE(3). Such systems emerge in many applications, such as quadrotor and quadcopter flight control. In the example, the paper demonstrates semi-global stabilization properties of the nonlinear MPC feedback law.

Model predictive control has been recognized as one of the essential solutions to achieve considerable energy savings in buildings. However, its performance on a building zone level can be inferior to a well-tuned conventional controller, especially in situations with constant energy prices and conservative comfort constraints. Optimization problem in the background has to be chosen to guarantee recursive feasibility and considerable energy savings without compromising the users comfort at the same time. This paper gives a novel formulation of the model predictive temperature control problem in buildings and its fair comparison with conventional controllers with the same level of flexibility allowed in zone temperature control. All controllers are tested for a system with seasonal heating and cooling, which is the most common case in real applications. It is shown that the introduced formulation leads to the model predictive controller that significantly outperforms conventional controllers both in energy consumption and users comfort.

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.

Adaptive MPC for Uncertain Discrete-Time LTI MIMO Systems with Incremental Input Constraints

This note proposes a model predictive control (MPC) algorithm for the solution of a robust control problem for continuous-time systems. Discontinuous feedback strategies are allowed in the solution of the min-max problems to be solved. The use of such strategies allows MPC to address a large class of nonlinear systems, including among others nonholonomic systems. Robust stability conditions to ensure steering to a certain set under bounded disturbances are established. The use of bang-bang feedbacks described by a small number of parameters is proposed, reducing considerably the computational burden associated with solving a differential game. The applicability of the proposed algorithm is tested to control a unicycle mobile robot.

This paper focuses on the model predictive control (MPC) issue where the measurable state is released under an event-triggered mechanism (ETM) to implement MPC over an infinite horizon. A new dynamic ETM (DETM) is devised to conserve network resources, which contains an additive internal dynamic variable (IDV), a multiplicative adaptively adjusting variable, a time-varying weighting matrix and several flexible scalars. The MPC problem is formulated as a ‘min–max’ optimisation problem (OP), where a hard constraint and robust positive invariant set on the predictive state/IDV are considered simultaneously. By resorting to a Lyapunov-like function that depends on the IDV, we put forward an auxiliary OP with matrix-inequality-based constraints. By the feasible solutions of such an auxiliary OP, the feedback gain matrix is designed which ensures the asymptotic stability of the closed-loop system. Two examples are presented to demonstrate the validity of the devised DETM and the DETM-based MPC algorithm. The study verifies that the devised DETM has advantages over an existing counterpart in conserving network resources while achieving the desired performance.

This paper aims to design a stabilized Adaptive Model Predictive Control (MPC) for nonlinear continuous systems with unknown dynamics at the start time. The proposed method is based on the Lyapunov MPC (LMPC) where the derivative of the Lyapunov function of the closed-loop system is limited to be less than that under an auxiliary Lyapunov controller. Different from the existing LMPC methods, this paper provides an online constraint updating strategy, which gradually reduces the conservation of the constraint. Specifically, the piece-wise linear (PWL) model updated by the Set-Membership (SM) identification method is adopted in the constraint and the auxiliary controller design, which can result in a non-increasing boundary of the model error. Then, a sufficient condition that guarantees the stabilization of the closed-loop system is deduced, which takes the effect of sample-and-hold implementation and the error boundary of the PWL model into account. By this condition, an optimization problem to update the Lyapunov controller for relaxing the constraints of MPC is proposed. We prove that, by the proposed method, the states will eventually converge to a small region around the equilibrium, both this small region and the conservatism of MPC decrease with the increasing of the accuracy of the PWL model. The application of the proposed method to a chemical process demonstrates the effectiveness of the proposed method. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —The control problem of nonlinear continuous systems with unknown dynamics has attracted the attention of researchers. This article develops an Adaptive MPC for this kind of system, which updates the boundary of uncertainties according to the measurement, and updates the constraints of MPC according to the updated uncertainty boundary. It provides a guaranteed stabilization and continuously relaxed constraint design for MPC where machine learning-based models are employed to predict the future trajectory. In the process of implementation, the method solves two optimization problems: one is to get the parameters of constraints which is based on the PWL model, and the other is used to optimize the future trajectory with the updated constraints. To proceed, the initial boundary of the uncertainty should be given at the initial instant, the initialization of PWL models and constraints can be configured based on it. The proposed method can be used for tracking control of chemical processes, power systems, robots, etc., where the accurate dynamics are not provided in advance and an improved performance is required.