Economic model predictive control for time-varying system: Performance and stability results

We consider economic model predictive control (MPC) without terminal conditions for time-varying optimal control problems. Under appropriate conditions, we prove that MPC yields initial pieces of approximately infinite horizon optimal trajectories and that the optimal infinite horizon trajectory is practically asymptotically stable. The results are illustrated by numerical examples motivated by energy-efficient heating and cooling of a building.     

Constructive model predictive control for constrained nonlinear systems

This paper develops a new model predictive control (MPC) design for stabilization of continuous‐time nonlinear systems subject to state and input constraints. The key idea is to construct an analytic form of the controller with some undetermined parameters and to calculate the parameters by minimizing online a performance index. By using the method of control Lyapunov functions (CLFs), we construct an appropriate variation on Sontag's formula, with one degree of freedom reflecting ‘decay rate’ of CLFs. Moreover, the constructed univariate control law is used to characterize the terminal region that guarantees the feasibility of the optimal control problem. Provided that the initial feasibility of the optimization problem is satisfied, the stability of the control scheme can be guaranteed. An example is given to illustrate the application of the constructive MPC design. Copyright © 2008 John Wiley & Sons, Ltd.     

Distributed model predictive control for continuous‐time nonlinear systems based on suboptimal ADMM

This paper presents a distributed model predictive control (DMPC) scheme for continuous‐time nonlinear systems based on the alternating direction method of multipliers (ADMM). A stopping criterion in the ADMM algorithm limits the iterations and therefore the required communication effort during the DMPC solution at the expense of a suboptimal solution. Stability results are presented for the suboptimal DMPC scheme under two different ADMM convergence assumptions. In particular, it is shown that the required iterations in each ADMM step are bounded, which is also confirmed in simulation studies.     

Stochastic optimal control problem for switching systems with constraints

This paper provides necessary conditions of optimality, in the form of a maximum principle, for optimal control problems of switching systems. Dynamics of the constituent processes take the form of stochastic differential equations with control terms in the drift and diffusion coefficients. The restrictions on the transitions or switches between operating modes are described by the collection of functional equalities. The main result is proved via an approximation functional and Ekeland's variational principle. Copyright © 2016 John Wiley & Sons, Ltd.     

LMI-based cooperative distributed model predictive control for Lipschitz nonlinear systems

In this paper, a distributed model predictive control is proposed to control Lipschitz nonlinear systems. The cooperative distributed scheme is considered where a global infinite horizon objective function is optimized for each subsystem, exploiting the state and input information of other subsystems. Thus, each control law is obtained separately as a state feedback of all system's states by solving a set of linear matrix inequalities. Due to convexity of the design, convergence properties at each iteration are established. Additionally, the proposed algorithm is modified to optimize only one control input at a time, which leads to a further reduction in the computation load. Finally, two application cases are studied to show the effectiveness of the proposed method.     

Investment portfolio tracking using model predictive control

The composition of an investment portfolio aiming to increase the financial returns while reducing the exposure to risk is a topic of growing interest in the world. In this direction, we propose a model predictive control (MPC) strategy in order to optimize the investment portfolio selection taking into account the tracking of a reference portfolio with desired return. In addition, an analysis comparing different sizes of the prediction horizon according to VPH-MPC (Varying Predictive Horizon-MPC) and FPH-MPC (Fixed Predictive Horizon-MPC) algorithms is conducted. Finally, we propose an optimal control problem using the tracking error as a function of loss of CVaR (Conditional Value at Risk) measurement. Numerical experiments are run based on Brazilian Stock Exchange data. The experimental results are compared with the Markowitz portfolio optimization model, a conventional tracking strategy, and benchmarks from the Brazilian financial market. This comparison indicates a good tracking performance obtained by the proposed MPC in the two versions while satisfying the constraints.     

Stochastic model predictive control for tracking linear systems

This note presents a stochastic formulation of the model predictive control for tracking (MPCT), based on the results of the work of Lorenzen et al. The proposed controller ensures constraints satisfaction in probability, and maintains the main features of the MPCT, that are feasibility for any changing setpoints and enlarged domain of attraction, even larger than the one delivered by Lorenzen et al, thanks to the use of artificial references and relaxed terminal constraints. The asymptotic stability (in probability) of the minimal robust positively invariant set centered on the desired setpoint is guaranteed. Simulations on a DC-DC converter show the benefits and the properties of the proposal.     

Switching optimal adaptive trajectory tracking control of quantum systems

In this paper, a switching optimal adaptive controller for tracking a time‐dependent trajectory in finite‐dimensional closed quantum systems is proposed. The issue of intrinsic singularities in trajectory tracking control of quantum systems leads to a sharp rise in the control amplitude. To overcome this drawback, a switching optimal adaptive quantum controller is designed based on Lyapunov stability theory and optimal quantum control approach. A state‐dependent strategy is considered to select the switching signal. The new switching controller adjusts the quantum state so that its population traces the desired dynamic trajectory and simultaneously eliminates the effects of singularities and reduces the control amplitude. The proposed controller is tested successfully for population transfer in a 4‐level closed quantum system in a simulation experiment. Both issues of reduction of the tracking error and control intensity along with a significant decrease in the number of singular points are well illustrated by simulation experiment as the advantages of the proposed method.     

Nonfragile predictive control for an omnidirectional rehabilitative training walker with constraints on the tracking errors of position and velocity

Here, we propose a novel nonfragile predictive control method for an omnidirectional rehabilitative training walker. The aim of the study was to design a tracking controller that includes simultaneous constraints on the tracking errors in the position and velocity in order to guarantee safe system states for the omnidirectional walker. The nonfragile controller uses incremental control to solve the optimal quadratic program problem. The exponential stability of the system is guaranteed by constructing constraints on the trajectory-tracking error and the velocity-tracking error. Simulation and experimental testing were conducted to confirm the effectiveness of the proposed method and verify that it can provide safe positions and velocities when applied to control the motion of an omnidirectional walker.     

Learning-based parametrized model predictive control for trajectory tracking

This article is concerned with the tracking of nonequilibrium motions with model predictive control (MPC). It proposes to parametrize input and state trajectories of a dynamic system with basis functions to alleviate the computational burden in MPC. As a result of the parametrization, an optimization problem with fewer variables is obtained, and the memory requirements for storing the reference trajectories are reduced. The article also discusses the generation of feasible reference trajectories that account for the system's dynamics, as well as input and state constraints. In order to cope with repeatable disturbances, which may stem from unmodeled dynamics for example, an iterative learning procedure is included. The approach relies on a Kalman filter that identifies the repeatable disturbances based on previous trials. These are then included in the system's model available to the model predictive controller, which compensates them in subsequent trials. The proposed approach is evaluated on a quadcopter, whose task is to balance a pole, while flying a predefined trajectory.     

Engine idle speed control using nonlinear multiparametric model predictive control

In this article, a real-time nonlinear model predictive idle speed controller based on multiparametric programming is designed for an SI engine. Idle speed is a crucial recurring condition in urban vehicles demanding proper control to avoid stall. As will be seen, the nonlinear model predictive control (NMPC) system designed, besides complying with the predefined constraints, demonstrates a far better performance than the prevalent industrial controllers and even conventional linear MPC controllers. More importantly, a new special structure combining offline nonlinear MPC and classical controller is employed to provide both robustness and fast response. Not only is the computational burden of the controller within that of the ordinary ECU controllers, it is also able to readily damp a disturbance of 20 N·m in less than 2.5 seconds and easily deal with parameter uncertainties. The control system also regulates engine gas pedal release, and converges to the set point with a settling time of less than 3 seconds and minimum fluctuations.     

Finite‐time convergence results in robust model predictive control

Robust asymptotic stability (asymptotic attractivity and ?‐δ stability) of equilibrium regions under robust model predictive control (MPC) strategies was extensively studied in the last decades making use of Lyapunov theory in most cases. However, in spite of its potential application benefits, the problem of finite‐time convergence under fixed prediction horizon has not received, with some few exceptions, much attention in the literature. Considering the importance in several applications of having finite‐time convergence results in the context of fixed horizon MPC controllers and the lack of studies on this matter, this work presents a new set‐based robust MPC (RMPC) for which, in addition to traditional stability guarantees, finite‐time convergence to a target set is proved, and moreover, an upper bound on the time necessary to reach that set is provided. It is remarkable that the results apply to general nonlinear systems and only require some weak assumptions on the model, cost function, and target set.     

Improving the grid frequency by optimal design of model predictive control with energy storage devices

This paper proposes the optimal design of model predictive control (MPC) with energy storage devices by the bat‐inspired algorithm (BIA) as a new artificial intelligence technique. Bat‐inspired algorithm‐based coordinated design of MPCs with superconducting magnetic energy storage (SMES) and capacitive energy storage (CES) is proposed for load frequency control. Three‐area hydrothermal interconnected power system installed with MPC and SMES is considered to carry out this study. The proposed design procedure can account for generation rate constraints and governor dead bands. Transport time delays imposed by governors, thermodynamic processes, and communication telemetry can be captured as well. In recent papers, the parameters of MPC with SMES and CES units are typically set by trial and error or by the designer's expertise. This problem is solved here by applying BIA to tune the parameters of MPC with SMES and CES units simultaneously to minimize the deviations of frequency and tie line powers against load perturbations. Simulation results are carried out to emphasize the superiority of the proposed coordinated design as compared with conventional proportional‐integral controller and with BIA‐based MPC without SMES and CES units.     

Resilient distributed model predictive control for energy management of interconnected microgrids

Distributed energy management of interconnected microgrids that is based on model predictive control (MPC) relies on the cooperation of all agents (microgrids). This paper discusses the case in which some of the agents might perform one type of adversarial actions (attacks) and they do not comply with the decisions computed by performing a distributed MPC algorithm. In this regard, these agents could obtain a better performance at the cost of degrading the performance of the network as a whole. A resilient distributed method that can deal with such issues is proposed in this paper. The method consists of two parts. The first part is to ensure that the decisions obtained from the algorithm are robustly feasible against most of the attacks with high confidence. In this part, we formulate the economic dispatch problem, taking into account the attacks as a chance-constrained problem, and employ a two-step randomization-based approach to obtain a feasible solution with a predefined level of confidence. The second part consists in the identification and mitigation of the adversarial agents, which utilizes hypothesis testing with Bayesian inference and requires each agent to solve a mixed-integer problem to decide the connections with its neighbors. In addition, an analysis of the decisions computed using the stochastic approach and the outcome of the identification and mitigation method is provided. The performance of the proposed approach is also shown through numerical simulations.     

Nonlinear model predictive control based on constraint transformation

The paper presents a constraint transformation approach for nonlinear model predictive control (MPC) subject to a class of state and control constraints. The approach uses a two‐stage transformation technique to incorporate the constraints into a new unconstrained MPC formulation with new variables. As part of the stability analysis, the relationship of the new unconstrained MPC scheme to an interior penalty formulation in the original variables is discussed. The approach is combined with an unconstrained gradient method that allows for computing the single MPC iterations in a real‐time manner. The applicability of the approach, for example, to fast mechatronic systems, is demonstrated by numerical as well as experimental results. Copyright © 2015 John Wiley & Sons, Ltd.     

A novel hybrid model predictive control design with application to a quadrotor helicopter

In this paper, a novel control design strategy based on a hybrid model predictive control in combination with fuzzy logic control is presented for a quadrotor helicopter system. In the proposed scheme, a 2‐part control structure is used. In the first part, a linear model predictive controller with receding horizon design strategy is combined with a nonlinear model predictive controller, which is applied as the main controller. In the second part, a 2‐level fuzzy logic controller is utilized to assist the first controller when the error exceeds a predefined value. The proposed nonlinear predictive control method utilizes a novel approach in which a prediction of the future outputs is used in the modeling stage. Using this simple technique, the problem can be solved using linear methods and, thereby, due to considerable reduction in the computational cost, it will be applicable for the systems with fast dynamics. Moreover, the fuzzy logic controller is used as a supervisor to adjust a proportional‐integral‐derivative controller to enhance the system performance by decreasing the tracking error. The proposed scheme is applied to a model of quadrotor system such that the difference between the predicted output of the system and the reference value is minimized while there are some constraints on inputs and outputs of the nonlinear quadrotor system. Simulation results demonstrate the efficiency of the proposed control scheme for the quadrotor system model.     

Inverse optimal control for asymptotic trajectory tracking of discrete‐time stochastic nonlinear systems in block controllable form

This paper concerns an inverse optimal control–based trajectory tracking of discrete‐time stochastic nonlinear systems. It is assumed that the nonlinear system can be transformed to the so called nonlinear block controllable form. Additionally, the synthesized control law minimizes a cost functional, which is posteriori determined. In contrast to the optimal control technique, this scheme avoids to solve the stochastic Hamilton‐Jacobi‐Bellman equation, which is not an easy task. Based on a discrete‐time stochastic control Lyapunov function, the proposed optimal controller is analyzed. The proposed approach is applied successfully to the two degrees‐of‐freedom helicopter with uncertainties in real time.     

A robust control system scheme based on model predictive controller (MPC) for continuous‐time systems

In this paper, an LMI framework based on model predictive strategy is addressed to design a robust dynamical control law in a typical control system. In the proposed method, instead of traditional static controller, a dynamic control law is used. With a suitable matrix transformation, the controller parameters selection are translated into an optimization problem with some LMI constraints. The plant input and output constraints are also handled with another LMIs. The controller is represented in state space form, and its parameters are computed in real‐time operation. For achieving this goal, by solving an optimization problem, a dynamic controller is designed, which meets the required plant performances. These results are used in 2 numerical examples to demonstrate the effectiveness of the proposed approach.     

Stable impulsive zone model predictive control for type 1 diabetic patients based on a long-term model

In this work, the problem of regulating blood glucose (glycemia) in type I diabetic patients is studied by means of an impulsive zone model predictive control (iZMPC), which bases its predictions on a novel long-term glucose-insulin model. Taking advantage of the impulsive version of the model—which features real-life properties of diabetes patients that some other popular models do not—the given control guarantees the stability under moderate-to-severe plant-model mismatch and disturbances. Long-term scenarios—including meals and physiological parameter variations—are simulated and the results are satisfactory as every hyperglycemic and hypoglycemic episodes are suitably controlled.     

Stochastic model predictive control approaches applied to drinking water networks

Control of drinking water networks is an arduous task, given their size and the presence of uncertainty in water demand. It is necessary to impose different constraints for ensuring a reliable water supply in the most economic and safe ways. To cope with uncertainty in system disturbances due to the stochastic water demand/consumption and optimize operational costs, this paper proposes three stochastic model predictive control (MPC) approaches, namely, chance‐constrained MPC, tree‐based MPC, and multiple‐scenario MPC. A comparative assessment of these approaches is performed when they are applied to real case studies, specifically, a sector and an aggregate version of the Barcelona drinking water network in Spain. Copyright © 2016 John Wiley & Sons, Ltd.