Robust Economic Model Predictive Control Research Articles

Reconfiguration of flow-based networks with back-up components using robust economic MPC

This paper addresses the post-fault selection of an actuators configuration for flow-based networks with back-up components. The proposed reconfiguration methodology consists of an offline and an online phase. On the one hand, an offline analysis looks for the minimal configurations for which the economic cost of the (best) steady-state trajectory that can be achieved using a robust model predictive control (MPC) policy is admissible. On the other hand, at fault detection time, an online search for the best actuators configuration to cope with the transient induced by the fault is conducted in the superset of each minimal configuration calculated offline. With this strategy, the final new configuration is computed by sequentially solving a set of mixed-integer programs whose constraints are derived from single-layer robust MPC schemes coupled with local controllers designed for the a priori minimal configurations identified offline. A portion of a water transport network is used to show the performance the proposed solution.

Optimal dispatching method for integrated energy system based on robust economic model predictive control considering source–load power interval prediction

Effective source–load prediction and reasonable dispatching are crucial to realize the economic and reliable operations of integrated energy systems (IESs). They can overcome the challenges introduced by the uncertainties of new energies and various types of loads in the IES. Accordingly, a robust optimal dispatching method for the IES based on a robust economic model predictive control (REMPC) strategy considering source–load power interval prediction is proposed. First, an operation model of the IES is established, and an interval prediction model based on the bidirectional long short-term memory network optimized by beetle antenna search and bootstrap is formulated and applied to predict the photovoltaic power and the cooling, heating, and electrical loads. Then, an optimal dispatching scheme based on REMPC is devised for the IES. The source–load interval prediction results are used to improve the robustness of the REPMC and reduce the influence of source–load uncertainties on dispatching. An actual IES case is selected to conduct simulations; the results show that compared with other prediction techniques, the proposed method has higher prediction interval coverage probability and prediction interval normalized averaged width. Moreover, the operational cost of the IES is decreased by the REMPC strategy. With the devised dispatching scheme, the ability of the IES to handle the dispatching risk caused by prediction errors is enhanced. Improved dispatching robustness and operational economy are also achieved.

Robust economic model predictive control with zone tracking

This paper presents a robust economic model predictive control (EMPC) formulation with zone tracking for discrete-time uncertain nonlinear systems. The proposed design ensures that the zone tracking objective is achieved in finite steps and at the same time optimizes the economic performance. In the proposed design, instead of tracking the original target zone, a robust control invariant set within the target zone is determined and is used as the actual zone tracked in the proposed EMPC. This approach ensures that the zone tracking objective is achieved within finite steps and once the zone tracking objective is achieved (the system state enters the robust control invariant set), the system state does not come out of the target zone anymore. To optimize the economic performance within the zone in the presence of disturbances, we introduce the notion of risk factor in the controller design. An algorithm to determine the economic zone to be tracked is provided. The risk factor determines the conservativeness of the controller and provides a way to tune the EMPC for better economic performance. A nonlinear chemical example is presented to demonstrate the performance of the proposed formulation.

Transient Performance of Tube-based Robust Economic Model Predictive Control

In this paper, we provide non-averaged and transient performance guarantees for recently developed, tube-based robust economic model predictive control (MPC) schemes. In particular, we consider both tube-based MPC schemes with and without terminal conditions. We show that the closed-loop performance obtained by applying such MPC schemes is approximately optimal when evaluated both on finite and infinite time horizons. These performance bounds are similar to those derived previously for nominal economic MPC. The theoretical results are discussed in a numerical example.

Robust Economic Model Predictive Control with Zone Control

Abstract This paper presents a robust economic Model Predictive Control (EMPC) formulation for discrete-time uncertain nonlinear systems. The proposed controller not only ensures that the closed-loop system is robust to disturbances, but also ensures that the economic performance does not deteriorate in the presence of the disturbances. The key idea is to have the controller track a robust control invariant subset of the state space with specified economic properties at all times, and within the zone optimize the process economics. To this end, we introduce the notion of risk factor in the controller design and provide an algorithm to determine the economic zone to be tracked. The risk factor determines the conservativeness of the controller. Our proposed controller is computationally less demanding as it only makes use of the system model without disturbances. A nonlinear CSTR example is presented to demonstrate the performance of the proposed formulation.

Economic model predictive control for robust optimal operation of sparse storage networks

A tube-based robust economic model predictive control scheme is proposed for sparse, linear storage network systems under bounded disturbances. We present a modeling framework that reflects conservation principles, and for which a centralized controller optimizes a performance metric for the entire network over a finite time horizon. Our main result is a computationally efficient projection-based approach that exploits the sparse structure of the system dynamics for tightening state and input constraints, in the meantime ensuring that average performance, recursive feasibility, and asymptotic stability of the system are maintained. An inventory control problem under demand uncertainty is presented to illustrate the advantages of the proposed framework and benchmark against nominal economic model predictive control.

A hybrid Gaussian process approach to robust economic model predictive control

This paper proposes a hybrid Gaussian process (GP) approach to robust economic model predictive control under unknown future disturbances in order to reduce the conservatism of the controller. The proposed hybrid GP is a combination of two well-known methods, namely, kernel composition and nonlinear auto-regressive. A switching mechanism is employed to select one of these methods for disturbance prediction after analyzing the prediction outcomes. The hybrid GP is intended to detect not only patterns but also unexpected behaviors in the unknown disturbances by using past disturbance measurements. A novel forgetting factor concept is also utilized in the hybrid GP, giving less weight to older measurements, in order to increase prediction accuracy based on recent disturbances values. The detected disturbance information is used to reduce prediction uncertainty in economic model predictive controllers systematically. The simulation results show that the proposed method can improve the overall performance of an economic model predictive controller compared to other GP-based methods in cases when disturbances have discernible patterns.

Homothetic tube-based robust offset-free economic Model Predictive Control

This paper proposes a novel economic Model Predictive Control algorithm aiming at achieving optimal steady-state performance despite the presence of plant-model mismatch or unmeasured nonzero mean disturbances. According to the offset-free formulation, the system’s state is augmented with disturbances and transformed into a new coordinate framework. Based on the new variables, the proposed controller integrates a moving horizon estimator to determine a solution of the nominal system surrounded by a set of potential states compatible with past input and output measurements. The worst cost within a single homothetic tube around the nominal solution is chosen as the economic objective function which is minimized to provide a tightened upper bound for the accumulated real cost within the prediction horizon window. Thanks to the combined use of the nominal system and homothetic tube, the designed optimization problem is recursively feasible and less conservative economic performance bounds are achieved. The proposed controller is demonstrated on a two-tanks system.

Robust economic model predictive control of nonlinear networked control systems with communication delays

SummaryIn this work, we consider economic model predictive control of nonlinear networked control systems subject to external disturbances and communication delays in both sensor‐to‐controller and controller‐to‐actuator channels. The problem is addressed in the framework of the min‐max model predictive control. First, a delay compensation strategy is proposed to minimize the impact of communication delays on the control performance. In the compensation strategy, once the receiver at the controller node receives a new state measurement, the controller generates a control sequence and sends the sequence to the actuator to compensate for delayed control inputs. Subsequently, the presence of disturbance is explicitly considered for robustness and the semi‐feedback min‐max optimization algorithm is used to design the control law based on the estimate of the current state reconstructed by the estimator. Furthermore, the input‐to‐state practical stability of the proposed approach is established by constructing a modified Lyapunov function. Simulation results of a numerical example and a chemical process example demonstrate the applicability and effectiveness of our approach.

Robust Economic Model Predictive Control Based on a Zonotope and Local Feedback Controller for Energy Dispatch in Smart-Grids Considering Demand Uncertainty

Electrical smart grids are complex MIMO systems whose operation can be noticeably affected by the presence of uncertainties such as load demand uncertainty. In this paper, based on a restricted representation of the demand uncertainty, we propose a robust economic model predictive control method that guarantees an optimal energy dispatch in a smart micro-grid. Load demands are uncertain, but viewed as bounded. The proposed method first decomposes control inputs into dependent and independent components, and then tackles the effect of demand uncertainty by tightening the system constraints as the uncertainty propagates along the prediction horizon using interval arithmetic and local state feedback control law. The tightened constraints’ upper and lower limits are computed off-line. The proposed method guarantees stability through a periodic terminal state constraint. The method is faster and simpler compared to other approaches based on Closed-loop min–max techniques. The applicability of the proposed approach is demonstrated using a smart micro-grid that comprises a wind generator, some photovoltaic (PV) panels, a diesel generator, a hydroelectric generator and some storage devices linked via two DC-buses, from which load demands can be adequately satisfied.

Robust economic model predictive control based on a periodicity constraint

SummaryThis paper proposes robust economic model predictive control based on a periodicity constraint for linear systems subject to unknown‐but‐bounded additive disturbances. In this economic MPC design, a periodic steady‐state trajectory is not required and thus assumed unknown, which precludes the use of enforcing terminal state constraints as in other standard economic formulations. Instead, based on the desired periodicity of system operation, we optimize the economic performance over a set of periodic trajectories that include the current state. To achieve robust constraint satisfaction, we use a tube‐based technique in the economic MPC formulation. The mismatches between the nominal model and the closed‐loop system with perturbations are limited using a local control law. With the proposed robust tube‐based strategy, recursive feasibility is guaranteed. Moreover, under a convexity assumption, the closed‐loop convergence of the closed‐loop system is analyzed, and an optimality certificate is provided to check if the closed‐loop trajectory reaches a neighborhood of the optimal nominal periodic steady trajectory using Karush‐Kuhn‐Tucker optimality conditions. Finally, through numerical examples, we show the effectiveness of the proposed approach.

Input‐to‐state stability of contractive EMPC of non‐linear systems with bounded disturbances

In this work, the authors consider the robust control problem of model predictive control (MPC) for economic criterion optimisation of constrained uncertain non-linear systems with unknown but bounded additive disturbances. The authors propose a robust economic MPC (EMPC) scheme with guaranteed input-to-state stability (ISS) to solve the robust control problem. By imposing a modified Lyapunov-based contractive constraint, the optimal control problem of the EMPC is formulated based on the nominal state prediction of the uncertain system. Then some sufficient conditions of Lipschitz continuity and terminal elements are derived to guarantee recursive feasibility and ISS of the EMPC with respect to the disturbance by the arguments of robust invariant sets and ISS. Two examples consisting of a chemical reaction process and a vehicle system are used to demonstrate the effectiveness of the proposed scheme.

Robust Contractive Economic MPC for Nonlinear Systems with Additive Disturbance

This article presents two novel robust contractive economic model predictive control (EMPC) algorithms with guaranteed input-to-state stability for nonlinear disturbed systems. Specifically, the first one adopts the non-quadratic economic objective function and implements the control sequence in block fashion, i.e., re-optimizing after all N control laws being applied. The second one takes receding horizon optimization strategy and only apply the first element in control sequence at each step. By appropriately chosen the contraction rate ρ in contractive constraint, the closed-loop system under each control algorithm is proven to be input-to-state stable. In order to illustrate the efficiency of proposed algorithms, a disturbed CSTR model is utilized.

Robust economic model predictive control: disturbance rejection, robustness and periodic operation in chemical reactors

ABSTRACTThis study investigates the properties of a robust economic model predictive control (REMPC) algorithm with respect to rejection of disturbances in initial conditions and non-stationary disturbances, robust stabilization and closed-loop performance in the presence of model parameters’ errors, and the enforcement of optimal economic periodic operations. A key characteristic of the algorithm is that it enforces robustness to model errors without requiring terminal conditions (unless periodic operation is desired) and instead a set-point trajectory is calculated at each time interval. Robust stability and convergence to the calculated set-point trajectory are enforced online by a set of constraints. Three case studies are used to illustrate the closed-loop performance of the REMPC algorithm for reactor operation under different conditions. The algorithm is shown to preserve stability in the presence of model parameters’ mismatch. In the face of disturbances, the algorithm leads to higher profits when a variable set-point is used compared to a fixed one. Furthermore, the periodic operation obtained through the application of the robust algorithm confers an improved average cost compared to steady-state operation.

A robust economic MPC for changing economic criterion

SummaryThis note presents a robust economic model predictive control controller suitable for changing economic criterion. The proposal ensures feasibility under any change of the economic criterion, thanks to the use of artificial variables and a relaxed terminal constraint, and robustness in presence of additive bounded disturbances. The resulting robust formulation considers a nominal prediction model and restricted constraints (in order to account for the effect of additive disturbances). The controlled system under the proposed controller is shown to be input‐to‐state stable in the sense that it is asymptotically steered to an invariant region around the best admissible steady state. An illustrative example shows the benefits and the properties of the proposed controller.

Robust Periodic Economic Model Predictive Control using Probabilistic Set Invariance for Descriptor Systems

Abstract This paper proposes a robust periodic economic model predictive control (MPC) using probabilistic set invariance for discrete-time descriptor systems. The considered systems are affected by additive disturbances that are bounded and follow a zero-mean Gaussian distribution. To guarantee robustness of the proposed robust economic MPC, a probabilistic invariant set with a given probability is used to contract the state and input constraints. An auxiliary input variable is defined to satisfy algebraic equations from descriptor systems in presence of disturbances. Finally, an application of the proposed MPC strategy to a smart micro-grid is provided to illustrate the performance.

Robust Periodic Economic Predictive Control based on Interval Arithmetic for Water Distribution Networks

This paper addresses a robust periodic economic model predictive control (EMPC) based on interval arithmetic with unknown-but-bounded additive disturbances for the management of water distribution networks (WDNs). The system constraints in presence of system disturbances are tightened along the prediction horizon by means of the proposed interval arithmetic and considering that system variables in the WDN model are also subject to some algebraic equations. These algebraic equations should be satisfied when the disturbances have effects on those system variables. The EMPC controller is designed with adding the terminal state constraint. The periodically optimal steady states are obtained by employing a periodic EMPC planner with the nominal model. This periodic steady states are subsequently used to be terminal states. The on-line robust constraint satisfactions are implemented into the robust periodic EMPC controller. Finally, the proposed control strategy is verified using a case study.

Economic Model Predictive Control and Applications for Diesel Generators

When developing control systems for diesel generators, tuning of the controller’s parameters to achieve acceptable performance is a significant challenge, particularly while satisfying input, emission, and safety constraints in the face of unknown system disturbances. Robust economic model predictive control (EMPC) can simplify this process by directly addressing the generator’s objectives, while systematically handling constraints in a robust way. This paper details how robust EMPC can be implemented as the control solution for diesel generators. To illustrate the process, two distinct generator applications are considered. The first application is a power tracking diesel generator, operating under emissions constraints. Such an application is found in series hybrid electric vehicles. The second application concerns diesel generators onboard submarines. In this application, engine speed and exhaust temperatures must be kept constant, despite significant system disturbances. An experimental study highlights the effectiveness of the EMPC as a solution for both applications.

Robust economic model predictive control of a community micro-grid

In this paper we propose a novel economic robust predictive controller for periodic operation. The proposed controller joins dynamic and economic trajectory planning and robust predictive controller for tracking in a single layer taking into account bounded disturbances, algebraic constraints and periodic operation. We study the closed-loop system properties of the proposed controller and provide a design procedure that guarantees that the perturbed closed-loop system converges asymptotically to the optimal economic reachable periodic trajectory, constraint satisfaction and recursive feasibility. The proposed controller has been applied to control a cluster of interconnected micro-grids. Each nano-grid is connected to an electric utility and has a renewable energy source, a cluster of batteries and a metal hydride based hydrogen storage system. The cluster must satisfy a periodic energy demand while maximizing the profit of the energy sold to the electric utility taking into account time varying prices.

Tube-based robust economic model predictive control

In this paper, we develop a tube-based economic MPC framework for nonlinear systems subject to unknown but bounded disturbances. Instead of simply transferring the design procedure of tube-based stabilizing MPC to an economic MPC framework, we rather propose to consider the influence of the disturbance explicitly within the design of the MPC controller, which can lead to an improved closed-loop average performance. This will be done by using a specifically defined integral stage cost, which is the key feature of our proposed robust economic MPC algorithm. Furthermore, we show that the algorithm enjoys similar properties as a nominal economic MPC algorithm (i.e., without disturbances), in particular with respect to bounds on the asymptotic average performance of the resulting closed-loop system, as well as stability and optimal steady-state operation.