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.     

A robust distributed model predictive control algorithm

Although distributed model predictive control (DMPC) has received significant attention in the literature, the robustness of DMPC with respect to model errors has not been explicitly addressed. In this paper, a novel online algorithm that deals explicitly with model errors for DMPC is proposed. The algorithm requires decomposing the entire system into N subsystems and solving N convex optimization problems to minimize an upper bound on a robust performance objective by using a time-varying state-feedback controller for each subsystem. Simulations examples were considered to illustrate the application of the proposed method.     

A switching robust model predictive control approach for nonlinear systems

This work considers enhancing the stability and improving the economic performance of nonlinear model predictive control in the presence of disturbances or model uncertainties. First, a robust control Lyapunov function (RCLF)-based predictive control strategy is proposed. Second, the approximate dynamic programming (ADP) is employed to further improve regulation performance. Finally, the ADP and RCLF-MPC are combined to provide a switching control scheme, which is illustrated on a CSTR example to show its effectiveness.     

Realization of robust nonlinear model predictive control by offline optimisation

This paper describes a new method for increasing the computational efficiency of nonlinear robust model-based predictive control. It is based on the application of neuro-fuzzy networks and improves the computation efficiency by arranging the online optimisation to be done offline. The offline optimisation is realized by offline training a neuro-fuzzy network, consisting of zero-order T–S fuzzy rules, which is designed to approximate the input–output relationship of a robust model-based predictive controller. The design and the training of the neuro-fuzzy network are described, and the corresponding control algorithm is developed. Experiment results performed on the temperature control loop of an experimental air-handling unit (AHU) demonstrate the effectiveness of this approach.     

基于多面体不变集的离线鲁棒预测控制器综合

提出一种基于多面体不变集的离线鲁棒预测控制器综合算法.该算法离线确定一组反馈控制律及其对应的不变集,在线控制时根据当前状态所处的位置选择相应的控制律,不仅扩大了初始可行域,还能在一定程度上改善控制性能.仿真结果表明了采用多面体不变集的优越性.     

On real-time robust model predictive control

High-speed applications impose a hard real-time constraint on the solution of a model predictive control (MPC) problem, which generally prevents the computation of the optimal control input. As a result, in most MPC implementations guarantees on feasibility and stability are sacrificed in order to achieve a real-time setting. In this paper we develop a real-time MPC approach for linear systems that provides these guarantees for arbitrary time constraints, allowing one to trade off computation time vs. performance. Stability is guaranteed by means of a constraint, enforcing that the resulting suboptimal MPC cost is a Lyapunov function. The key is then to guarantee feasibility in real-time, which is achieved by the proposed algorithm through a warm-starting technique in combination with robust MPC design. We address both regulation and tracking of piecewise constant references. As a main contribution of this paper, a new warm-start procedure together with a Lyapunov function for real-time tracking is presented. In addition to providing strong theoretical guarantees, the proposed method can be implemented at high sampling rates. Simulation examples demonstrate the effectiveness of the real-time scheme and show that computation times in the millisecond range can be achieved.     

Efficient algorithms for online tracking of set points in robust model predictive control

This paper presents some computationally efficient algorithms for online tracking of set points in robust model predictive control context subject to state and input constraints. The nonlinear systems are represented by a linear model along with an additive nonlinear term which is locally Lipschitz. As an unstructured uncertainty, this term is replaced in the robust stability constraint by its Lipschitz coefficient. A scheduled control technique is employed to transfer the system to desired set points, given online, by designing local robust model predictive controllers. This scheme includes estimating the regions of feasibility and stability of the related equilibriums and online switching among the local controllers. The proposed optimisation problems for calculating the regions of feasibility and stability are defined as linear matrix inequalities that can be solved in polynomial time. The effectiveness of the proposed algorithms is illustrated by an example.     

On the explicit dead-time compensation for robust model predictive control

Computational simplicity is one of the most important aspects to take into account in robust model predictive control (MPC). In dead-time processes, it is common to use an augmented state-space representation in order to apply robust MPC strategies but, this procedure may affect computational aspects. In this paper, explicit dead-time compensation will be used to avoid augmented representation. This technique will be analyzed in terms of robust stability and constraint satisfaction for discrete-time linear systems. The results of this discussion will be applied to a robust tube-based MPC strategy which is able to guarantee robust stability and constraint satisfaction of a dead-time system by considering a prediction model without dead-time. Moreover, taking advantage of the proposed scheme, the robust MPC will be particularized for first-order plus dead-time models which simplifies significantly controller synthesis. The proposed dead-time compensation method will be applied to different robust MPC strategies in two case studies: (i) a simulated quadruple-tank system, and (ii) an experimental scaled laboratory heater process.     

Disturbance modeling for offset-free linear model predictive control

An offset-free controller is one that drives controlled outputs to their desired targets at steady state. In the linear model predictive control (MPC) framework, offset-free control is usually achieved by adding step disturbances to the process model. The most widely-used industrial MPC implementations assume a constant output disturbance that can lead to sluggish rejection of disturbances that enter the process elsewhere. This paper presents a general disturbance model that accommodates unmeasured disturbances entering through the process input, state, or output. Conditions that guarantee detectability of the augmented system model are provided, and a steady-state target calculation is constructed to remove the effects of estimated disturbances. Conditions for which offset-free control is possible are stated for the combined estimator, steady-state target calculation, and dynamic controller. Simulation examples are provided to illustrate trade-offs in disturbance model design.     

New offset-free method for model predictive control of open channels

Irrigation or drainage canals can be controlled by model predictive control (MPC). Applying MPC with an internal model in the presence of unknown disturbances in some cases can lead to steady state offset. Therefore an additional component should be implemented along with the MPC. A new method eliminating the offset has been developed in this paper for MPC. It is based on combining two basic approaches of MPC. It has been implemented to control water levels in the three-pool UPC laboratory canal and further numerically tested using a test case benchmark proposed by the American Society of Civil Engineers (ASCE). It has been found that the developed offset-free method is able to eliminate the steady-state offset, while taking into account known and unknown disturbances.     

Handling uncertainty in economic nonlinear model predictive control: A comparative case study

In the last years, the use of an economic cost function for model predictive control (MPC) has been widely discussed in the literature. The main motivation for this choice is that often the real goal of control is to maximize the profit or the efficiency of a certain system, rather than tracking a predefined set-point as done in the typical MPC approaches, which can be even counter-productive. Since the economic optimal operation of a system resulting from the application of an economic model predictive control approach drives the system to the constraints, the explicit consideration of the uncertainties becomes crucial in order to avoid constraint violations. Although robust MPC has been studied during the past years, little attention has yet been devoted to this topic in the context of economic nonlinear model predictive control, especially when analyzing the performance of the different MPC approaches. In this work, we present the use of multi-stage scenario-based nonlinear model predictive control as a promising strategy to deal with uncertainties in the context of economic NMPC. We make a comparison based on simulations of the advantages of the proposed approach with an open-loop NMPC controller in which no feedback is introduced in the prediction and with an NMPC controller which optimizes over affine control policies. The approach is efficiently implemented using CasADi, which makes it possible to achieve real-time computations for an industrial batch polymerization reactor model provided by BASF SE. Finally, a novel algorithm inspired by tube-based MPC is proposed in order to achieve a trade-off between the variability of the controlled system and the economic performance under uncertainty. Simulations results show that a closed-loop approach for robust NMPC increases the performance and that enforcing low variability under uncertainty of the controlled system might result in a big performance loss.     

Least-restrictive robust periodic model predictive control applied to room temperature regulation

State-feedback model predictive control (MPC) of constrained discrete-time periodic affine systems is considered. The periodic systems’ states and inputs are subject to periodically time-dependent, hard, polyhedral constraints. Disturbances are additive, bounded and subject to periodically time-dependent bounds. The objective is to design MPC laws that robustly enforce constraint satisfaction in a manner that is least-restrictive, i.e., have the largest possible domain. The proposed design method is demonstrated on a building climate control example. The proposed method is directly applicable to time-invariant MPC.     

采用LMI的鲁棒预测控制综合方法

本文简单叙述了线性矩阵不等式（LMI）在控制理论中应用的历史,重点介绍了LMI的原理及其在鲁棒预测控制综合方法中应用的一种新方法,举例说明了如何在预测控制综合方法中应用LMI。本文可作为在鲁棒预测控制中应用LMI和用微机实现鲁棒预测控制算法的参考。     

A robust/cautious model predictive controller

In process control, a significant number of problems are encountered where there are hard and soft constraints on the measured process variables and/or on the value and rate of change of the manipulated variables. On-line implementation of traditional control strategies becomes unfeasible since they cannot explicitly deal with process constraints. Model predictive control offers a viable alternative; however, a prominent issue is the behaviour of these algorithms when the prediction model does not match the actual plant. A possible solution is to formulate the problem in a linear/non-linear programming framework, using the cutting plane technique to locate the ‘worst’ plant/model mismatch at every time interval. This results in a very practical cautious predictive controller that computes the next controller action based on expected model/plant mismatch. Control of a stirred tank reactor illustrates the method, using a one-step ahead predictive controller.     

Event-triggered robust model predictive control of continuous-time nonlinear systems

The event-triggered control is of compelling features in efficiently exploiting system resources, and thus has found many applications in sensor networks, networked control systems, multi-agent systems and so on. In this paper, we study the event-triggered model predictive control (MPC) problem for continuous-time nonlinear systems subject to bounded disturbances. An event-triggered mechanism is first designed by measuring the error between the system state and its optimal prediction; the event-triggered MPC algorithm that is built upon the triggering mechanism and the dual-mode approach is then designed. The rigorous analysis of the feasibility and stability is conducted, and the sufficient conditions for ensuring the feasibility and stability are developed. We show that the feasibility of the event-triggered MPC algorithm can be guaranteed if, the prediction horizon is designed properly and the disturbances are small enough. Furthermore, it is shown that the stability is related to the prediction horizon, the disturbance bound and the triggering level, and that the state trajectory converges to a robust invariant set under the proposed conditions. Finally, a case study is provided to verify the theoretical results.     

Causal state-feedback parameterizations in robust model predictive control

In this paper, we investigate the problem of nonlinearity (and non-convexity) typically associated with linear state-feedback parameterizations in the Robust Model Predictive Control (RMPC) for uncertain systems. In particular, we propose two tractable approaches to compute an RMPC controller–consisting of both a causal, state-feedback gain and a control-perturbation component–for linear, discrete-time systems involving bounded disturbances and norm-bounded structured model-uncertainties along with hard constraints on the input and state. Both the state-feedback gain and the control-perturbation are explicitly considered as decision variables in the online optimization while avoiding nonlinearity and non-convexity in the formulation. The proposed RMPC controller–computed through LMI optimizations–is responsible for steering the uncertain system state to a terminal invariant set. Numerical examples from the literature demonstrate the advantages of the proposed scheme.     

Tube‐based robust nonlinear model predictive control

This paper extends tube‐based model predictive control of linear systems to achieve robust control of nonlinear systems subject to additive disturbances. A central or reference trajectory is determined by solving a nominal optimal control problem. The local linear controller, employed in tube‐based robust control of linear systems, is replaced by an ancillary model predictive controller that forces the trajectories of the disturbed system to lie in a tube whose center is the reference trajectory thereby enabling robust control of uncertain nonlinear systems to be achieved. Copyright © 2011 John Wiley & Sons, Ltd.     

Designing of non-fragile robust model predictive control for constrained uncertain systems and its application in process control

A non-fragile robust model predictive control (RMPC) is designed in the uncertain systems under bounded control signals. To this aim, a class of the nonlinear systems with additive uncertainty is considered in its general form. The RMPC synthesis could lead to the proper selection of the controller’s gains. Thus, the non-fragile RMPC design is translated into a minimization problem subjected to some constraints in terms of linear matrix inequality (LMI). Hence, the controller’s gains are computed by solving such a minimization problem. In some numerical examples, the suggested non-fragile RMPC is compared with the other methods. The simulation results demonstrate the effectiveness of the proposed RMPC in comparison with similar techniques.