Plug-and-play model predictive control based on robust control invariant sets

We consider the problem of designing decentralized controllers for large-scale linear constrained systems composed by a number of interacting subsystems. As in Riverso et al. (2013b), (i) the design of local controllers requires limited transmission of information from other subsystems and (ii) the addition/removal of a subsystem triggers the design of local controllers for child subsystems only. These properties enable Plug-and-Play (PnP) operations, and we show how to perform them while preserving global stability of the origin and constraint satisfaction. We improve several aspects of the PnP design procedure proposed in Riverso et al. (2013b) and, using recent results in the computation of Robust Control Invariant (RCI) sets, we show that all critical steps in the design of a local controller can be solved through Linear Programming (LP). Finally, an application of the proposed design procedure to a large-scale mechanical system is presented.     

Constrained robust model predictive control based on periodic invariance

In this paper, a receding-horizon control method for input/state constrained systems with polyhedral uncertainties is proposed. The dual-mode prediction strategy is adopted to deal with the constraints and periodically-invariant sets are used to derive a target invariant set of the dual-mode prediction strategy. The proposed control method is shown to have novel characteristics earlier approaches do not have i.e.: (i) the convex-hull of all the periodically invariant sets are invariant in the sense that there are feasible feedback gains guaranteeing invariance for any elements of the convex-hull and it provides larger target sets than other methods based on ordinary invariant sets. (ii) A particular convex-hull of periodically invariant sets, that is computable off-line, can be used as an invariant target set. In this case the number of on-line variables is only equal to the period of invariance and thus the proposed algorithm is computationally very efficient. These on-line variables provide interpolation between different feedback gains to yield best performance.     

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.     

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.     

Nonlinear system modeling and robust predictive control based on RBF-ARX model

An integrated modeling and robust model predictive control (MPC) approach is proposed for a class of nonlinear systems with unknown steady state. First, the nonlinear system is identified off-line by RBF-ARX model possessing linear ARX model structure and state-dependent Gaussian RBF neural network type coefficients. On the basis of the RBF-ARX model, a combination of a local linearization model and a polytopic uncertain linear parameter-varying (LPV) model are built to approximate the present and the future system's nonlinear behavior, respectively. Subsequently, based on the approximate models, a min–max robust MPC algorithm with input constraint is designed for the output-tracking control of the nonlinear system with unknown steady state. The closed-loop stability of the MPC strategy is guaranteed by the use of parameter-dependent Lyapunov function and the feasibility of the linear matrix inequalities (LMIs). Simulation study to a NO_x decomposition process illustrates the effectiveness of the modeling and robust MPC approaches proposed in this paper.     

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.     

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.     

基于鲁棒控制Lyapunov 函数的非线性预测控制

针对一类约束不确定性非线性仿射系统,提出一种可保证闭环系统鲁棒镇定的非线性模型预测控制算法.利用鲁棒控制Lyapunov函数得到改进的Sontag公式,并以此为基础,构造一种计算有效的单自由度鲁棒预测控制器.以Matlab语言为仿真工具,对一开环不稳定振荡器进行了仿真研究,结果表明,利用该控制算法得到的闭环系统不仅渐近稳定于原点,而且所得控制量和系统状态都满足系统约束,从而验证了控制算法的有效性.     

基于LMI的多模型鲁棒预测控制

用线性矩阵不等式 (LMI)方法研究多模型鲁棒预测控制, 提出了状态反馈的综合方法, 并分析了闭环系统的可行性, 同时证明闭环系统渐近稳定. 在此基础上, 研究了带终端零状态的有限优化时域预测控制和无穷优化时域预测控制的性能, 证明了两者在性能上的一致性.     

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.     

Dynamic output feedback robust model predictive control

The synthesis approach for dynamic output feedback robust model predictive control is considered. The notion of quadratic boundedness is utilised to characterise the stability properties of the augmented closed-loop system. A finite horizon performance cost, which corresponds to the worst case of both the polytopic uncertainty and the bounded disturbance/noise, is utilised. It is not required to specify the horizon length. A numerical example is given to illustrate the effectiveness of the proposed controller.     

基于状态观测器的直接约束鲁棒预测控制

针对多包描述线性离散不确定系统,提出一种在系统状态不可测时的直接约束鲁棒预测控制算法.将控制器与观测器综合设计,利用观测状态直接构造性能指标,通过求解无穷时域性能指标的最小最大优化问题,得到系统的最优状态反馈控制律.采用参数依赖Lyapunov函数,在满足输入和状态约束的情况下保证闭环系统稳定.仿真结果验证了算法的有效性.     

On robust tracking with non-linear model predictive control

In this paper a class of stabilizing model predictive control algorithms based on an optimization problem with two different horizons (control and prediction) is considered. Particular attention is devoted to the tracking problem for reference signals constant beyond a prescribed future horizon. For such a problem, it is pointed out the necessity of an output-feedback scheme to guarantee robust steady-state zero-error regulation. Moreover, an additional constraint is introduced in the optimization problem for the robust fulfilment of the constraints on the control variable. The potentiality of this approach is shown through some simulation experiments on a highly non-linear chemical reactor.     

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.     

Improved robust model predictive control with structured uncertainty

In this paper, a dilation of the LMI characterization is presented to address constrained robust model predictive control (MPC) for a class of uncertain linear systems with structured time-varying uncertainties. The uncertainty is described in linear fractional transformation (LFT) form. By introducing slack variables and using parameter dependent Lyapunov functions, the design conservativeness is reduced compared with other existing MPC approaches. The proposed approach is applied to an industrial CSTR benchmark system to demonstrate the merits of our proposed solution.     

Application of interval predictor model into robust model predictive control

Here we apply interval prediction model into robust model predictive control (MPC) strategy. After introducing the family of models and some basic information, we present the computational results for the construction of interval predictor model, whose regression parameters are included in a sphere parameter set. Given a size measure to scale the average amplitude of the predictor interval, one optimal model that minimises a size measure is efficiently computed by solving a linear programming problem. We apply the active set approach to solve the linear programming problem and based on these optimisation variables, the predictor interval of the considered model with sphere parameter set can be constructed. As for a fixed non-negative number from the size measure, we propose a better choice by using the optimality conditions. In order to apply interval prediction model into robust MPC, two strategies are proposed to analyse a min-max optimisation problem. After input and output variables are regarded as decision variables, a standard quadratic optimisation is obtained and its dual form is derived, then Gauss–Seidel algorithm is proposed to solve the dual problem and convergence of Gauss–Seidel algorithm is provided too. Finally two simulation examples confirm the theoretical results.     

Efficient robust predictive control

Predictive constrained control of time-varying and/or uncertain linear systems has been effected through the use of ellipsoidal invariant sets (Kothare et al., 1996). Linear matrix inequalities (LMIs) have been used to design a state-dependent state-feedback law that maintains the state vector inside invariant feasible sets. For the purposes of prediction however, at each time instant, the state feedback law is assumed constant. In addition, due to the large number of LMIs involved, online computation becomes intractable for anything other than small dimensional systems. Here we propose an approach that deploys a fixed state-feedback law but introduces extra degrees of freedom through the use of perturbations on the fixed state-feedback law. The problem is so formulated that all demanding computations can be performed offline leaving only a simple optimization problem to be solved online. Over and above the very significant reduction in computational cost, the extra degrees of freedom allow for better performance and wider applicability     

Analytical performance prediction for robust constrained model predictive control

This paper presents a new analysis tool for predicting the closed-loop performance of a robust constrained model predictive control (MPC) scheme. Currently, performance is typically evaluated by numerical simulation, leading to an extensive computation when investigating the effect of controller parameters, such as the horizon length, the cost weightings and the constraint settings. The analytic method, in this paper, avoids this computational burden, thus enabling a rapid study of the trades between the design parameters and the performance. Previous work developed an MPC formulation employing constraint tightening to achieve robust feasibility and constraint satisfaction despite the action of an unknown but bounded disturbance. This paper shows that the expected performance of that controller can be predicted using a combination of the gains of two linear systems, the optimal control for the unconstrained system, and a candidate policy used in performing the constraint tightening. The method also accounts for the possible mismatch between the predicted level of disturbance and the actual level encountered. The analytic results are compared with simulation results for several examples and are shown to provide accurate predictions of performance and its variation with the system parameters.     

离线鲁棒预测控制器综合方法的改进方案

对离线鲁棒预测控制器综合方法进行改进.离线鲁棒预测控制算法离线确定一个控制律序列,对应一组吸引域;在线根据当前状态位于哪个吸引域内部(或哪两个吸引域之间)选择相应的一个控制律(或相邻的两个控制律的线性插值).引入参数Lyapunov函数得到多包椭圆型吸引域和多包二次稳定的离线控制律,改进了控制器的可行性和最优性.改进的控制器保持了稳定性以及控制律关于系统状态的连续性.     

Nash-based robust distributed model predictive control for large-scale systems

In this paper, a new robust distributed model predictive control (RDMPC) is proposed for large-scale systems with polytopic uncertainties. The time-varying system is first decomposed into several interconnected subsystems. Interactions between subsystems are obtained by a distributed Kalman filter, in which unknown parameters of the system are estimated using local measurements and measurements of neighboring subsystems that are available via a network. Quadratic boundedness is used to guarantee the stability of the closed-loop system. In the MPC algorithm, an output feedback-interaction feedforward control input is computed by an LMI-based optimization problem that minimizes an upper bound on the worst case value of an infinite-horizon objective function. Then, an iterative Nash-based algorithm is presented to achieve the overall optimal solution of the whole system in partially distributed fashion. Finally, the proposed distributed MPC approach is applied to a load frequency control (LFC) problem of a multi-area power network to study the efficiency and applicability of the algorithm in comparison with the centralized, distributed and decentralized MPC schemes.