Robust MPC for tracking zone regions based on nominal predictions

This paper deals with the problem of robust tracking of target sets using a model predictive control (MPC) law. Real industries applications often require a control strategy in which some system outputs are controlled within specified ranges or zones (zone control), while some others variables – possibly including input variables – are steered to fixed target or setpoint. From a theoretical point of view, the control objective of this kind of problem can be seen as a target set (in the output space) instead of a target point, since inside the zones there are no preferences between one point or another. This problem is particularly interesting in case of additive disturbances which might push the outputs out of the zones. In this work, a stable robust MPC formulation for constrained linear systems, based on nominal predictions is presented. The main features of this controller are the use of nominal predictions, restricted constraints and the concept of distance from a point to a set as offset cost function. The controller ensures both recursive feasibility and local optimality. The properties of the controller are shown in a simulation test, in which we consider a subsystem of an industrial FCC system.     

A stable MPC with zone control

Several MPC applications implement a control strategy in which some of the system outputs are controlled within specified ranges or zones, rather than at fixed set points [J.M. Maciejowski, Predictive Control with Constraints, Prentice Hall, New Jersey, 2002]. This means that these outputs will be treated as controlled variables only when the predicted future values lie outside the boundary of their corresponding zones. The zone control is usually implemented by selecting an appropriate weighting matrix for the output error in the control cost function. When an output prediction is inside its zone, the corresponding weight is zeroed, so that the controller ignores this output. When the output prediction lies outside the zone, the error weight is made equal to a specified value and the distance between the output prediction and the boundary of the zone is minimized. The main problem of this approach, as long as stability of the closed loop is concerned, is that each time an output is switched from the status of non-controlled to the status of controlled, or vice versa, a different linear controller is activated. Thus, throughout the continuous operation of the process, the control system keeps switching from one controller to another. Even if a stabilizing control law is developed for each of the control configurations, switching among stable controllers not necessarily produces a stable closed loop system.Here, a stable MPC is developed for the zone control of open-loop stable systems. Focusing on the practical application of the proposed controller, it is assumed that in the control structure of the process system there is an upper optimization layer that defines optimal targets to the system inputs. The performance of the proposed strategy is illustrated by simulation of a subsystem of an industrial FCC system.     

MPC for tracking piecewise constant references for constrained linear systems

In this paper, a novel model predictive control (MPC) for constrained (non-square) linear systems to track piecewise constant references is presented. This controller ensures constraint satisfaction and asymptotic evolution of the system to any target which is an admissible steady-state. Therefore, any sequence of piecewise admissible setpoints can be tracked without error. If the target steady state is not admissible, the controller steers the system to the closest admissible steady state.These objectives are achieved by: (i) adding an artificial steady state and input as decision variables, (ii) using a modified cost function to penalize the distance from the artificial to the target steady state (iii) considering an extended terminal constraint based on the notion of invariant set for tracking. The control law is derived from the solution of a single quadratic programming problem which is feasible for any target. Furthermore, the proposed controller provides a larger domain of attraction (for a given control horizon) than the standard MPC and can be explicitly computed by means of multiparametric programming tools. On the other hand, the extra degrees of freedom added to the MPC may cause a loss of optimality that can be arbitrarily reduced by an appropriate weighting of the offset cost term.     

一类干扰有界约束非线性系统的鲁棒模型预测控制

针对一类干扰有界约束非线性系统设计了基于控制不变集切换策略的鲁棒模型预测控制算法.针对非线性系统线性化之后的结果,给出了平衡点的非线性标称系统控制不变集的计算方法.然后在考虑线性化误差和加性有界干扰影响的基础上,构造了平衡点附近最小鲁棒正不变集.结合不变集切换策略和Tube不变集控制方法,提出了干扰有界约束非线性系统的不变集切换策略.最后将该算法应用到一类典型的非线性化工过程连续搅拌反应釜(CSTR)中,仿真结果验证了算法的有效性.     

Stochastic tube MPC with state estimation

An output feedback Model Predictive Control (MPC) strategy for linear systems with additive stochastic disturbances and probabilistic constraints is proposed. Given the probability distributions of the disturbance input, the measurement noise and the initial state estimation error, the distributions of future realizations of the constrained variables are predicted using the dynamics of the plant and a linear state estimator. From these distributions, a set of deterministic constraints is computed for the predictions of a nominal model. The constraints are incorporated in a receding horizon optimization of an expected quadratic cost, which is formulated as a quadratic program. The constraints are constructed so as to provide a guarantee of recursive feasibility, and the closed loop system is stable in a mean-square sense. All uncertainties in this paper are taken to be bounded—in most control applications this gives a more realistic representation of process and measurement noise than the more traditional Gaussian assumption.     

Non-fragile observer-based output feedback control for polytopic uncertain system under distributed model predictive control approach

In this paper, a non-fragile observer-based output feedback control problem for the polytopic uncertain system under distributed model predictive control (MPC) approach is discussed. By decomposing the global system into some subsystems, the computation complexity is reduced, so it follows that the online designing time can be saved.Moreover, an observer-based output feedback control algorithm is proposed in the framework of distributed MPC to deal with the difficulties in obtaining the states measurements. In this way, the presented observer-based output-feedback MPC strategy is more flexible and applicable in practice than the traditional state-feedback one. What is more, the non-fragility of the controller has been taken into consideration in favour of increasing the robustness of the polytopic uncertain system. After that, a sufficient stability criterion is presented by using Lyapunov-like functional approach, meanwhile, the corresponding control law and the upper bound of the quadratic cost function are derived by solving an optimisation subject to convex constraints. Finally, some simulation examples are employed to show the effectiveness of the method.     

MPC for tracking with optimal closed-loop performance

In the recent paper [Limon, D., Alvarado, I., Alamo, T., & Camacho, E.F. (2008). MPC for tracking of piece-wise constant references for constrained linear systems. Automatica, 44, 2382-2387], a novel predictive control technique for tracking changing target operating points has been proposed. Asymptotic stability of any admissible equilibrium point is achieved by adding an artificial steady state and input as decision variables, specializing the terminal conditions and adding an offset cost function to the functional.In this paper, the closed-loop performance of this controller is studied and it is demonstrated that the offset cost function plays an important role in the performance of the model predictive control (MPC) for tracking. Firstly, the controller formulation has been enhanced by considering a convex, positive definite and subdifferential function as the offset cost function. Then it is demonstrated that this formulation ensures convergence to an equilibrium point which minimizes the offset cost function. Thus, in case of target operation points which are not reachable steady states or inputs for the constrained system, the proposed control law steers the system to an admissible steady state (different to the target) which is optimal with relation to the offset cost function. Therefore, the offset cost function plays the role of a steady-state target optimizer which is built into the controller. On the other hand, optimal performance of the MPC for tracking is studied and it is demonstrated that under some conditions on both the offset and the terminal cost functions optimal closed-loop performance is locally achieved.     

约束非线性系统切换鲁棒预测控制

采用“分段蕴含”(PWE)方法, 用一组线性变参数模型(LPV)近似约束非线性系统, 降低模型近似的保守性. 对每个LPV模型引入参数Lyapunov函数, 得到稳定的控制律, 并施加于非线性系统. 当检测到LPV模型发生切换时, 根据可行域的离线设计方法确定适当的切换律, 使系统按照设定的规则切换, 保证切换后的初始状态可行. 在文章最后给出了基于切换策略的控制算法的可行性和稳定性. 与传统非线性预测控制相比, 基于切换策略的鲁棒预测 控制方法保守性更低, 计算量更小.     

Distributed model predictive control for constrained linear systems

We address the distributed model predictive control (MPC) for a set of linear local systems with decoupled dynamics and a coupled global cost function. By the decomposition of the global cost function, the distributed control problem is converted to the MPC for each local system associated with a cost involving neighboring system states and inputs. For each local controller, the infinite horizon control moves are parameterized as N free control moves followed by a single state feedback law. An interacting compatibility condition is derived, disassembled and incorporated into the design of each local control so as to achieve the stability of the global closed‐loop system. Each local system exchanges with its neighbors the current states and the previous optimal control strategies. The global closed‐loop system is shown to be exponentially stable provided that all the local optimizers are feasible at the initial time. Copyright © 2009 John Wiley & Sons, Ltd.     

Robust tube-based MPC for tracking of constrained linear systems with additive disturbances

This paper is devoted to solve the problem that the predictive controllers may present when the target operation point changes. Model predictive controllers (MPC) are capable to steer an uncertain system to a given target operation point fulfilling the constraints. But if the target changes significantly the controller may not success due to the loss of feasibility of the optimization problem and the inadequacy of the terminal conditions.This paper presents a novel formulation of a robust model predictive controller (MPC) for tracking changing targets based on a single optimization problem. The plant is assumed to be modelled as a linear system with additive uncertainties confined to a bounded known polyhedral set. Under mild assumptions, the proposed MPC is feasible under any change of the target and steers the uncertain system to (a neighborhood of) the target if this is admissible. If the target is not admissible, and hence unreachable, the system is steered to the closest admissible operating point.The controller formulation has some parameters which provide extra degrees of freedom. These new parameters allow control objectives such as disturbance rejection, output offset prioritization or enlargement of the domain of attraction to be dealt with. The paper shows how these parameters can be calculated off-line.In order to demonstrate the benefits of the proposed controller, it has been tested on a real plant: the four tanks plant which is a multivariable nonlinear system configured to exhibit non-minimum phase transmission zeros. Experimental results show the robust stability and offset-free tracking of the controlled plant.     

一类非线性不确定时滞系统的混杂状态反馈H∞鲁棒控制

利用混杂状态反馈控制策略研究一类非线性不确定时滞系统的H∞鲁棒控制问题.假定在给定的控制器集合中有有限个备选的状态反馈控制器,并且每个单一的连续控制器都不能使系统具有鲁棒H∞性能.当控制器的增益矩阵已知时,基于单Lyapunov函数技术和凸组合条件给出控制器切换方案以确保非线性不确定时滞系统具有鲁棒H∞性能.当控制器的增益矩阵未知时,使用多Lyapunov函数技术得到了问题可解的另一个充分条件,同时还给出了混杂状态反馈H∞控制器的设计.     

Stochastic model predictive control for probabilistically constrained Markovian jump linear systems with additive disturbance

Model predictive control (MPC) for Markovian jump linear systems with probabilistic constraints has received much attention in recent years. However, in existing results, the disturbance is usually assumed with infinite support, which is not considered reasonable in real applications. Thus, by considering random additive disturbance with finite support, this paper is devoted to a systematic approach to stochastic MPC for Markovian jump linear systems with probabilistic constraints. The adopted MPC law is parameterized by a mode‐dependent feedback control law superimposed with a perturbation generated by a dynamic controller. Probabilistic constraints can be guaranteed by confining the augmented system state to a maximal admissible set. Then, the MPC algorithm is given in the form of linearly constrained quadratic programming problems by optimizing the infinite sum of derivation of the stage cost from its steady‐state value. The proposed algorithm is proved to be recursively feasible and to guarantee constraints satisfaction, and the closed‐loop long‐run average cost is not more than that of the unconstrained closed‐loop system with static feedback. Finally, when adopting the optimal feedback gains in the predictive control law, the resulting MPC algorithm has been proved to converge in the mean square sense to the optimal control. A numerical example is given to verify the efficiency of the proposed results.     

Switched model predictive control for performance enhancement

This article presents a switched model predictive control (MPC) algorithm for non-linear discrete time systems where the weights on the state and control variables in the cost function to be minimised depend on the current value of the state. In so doing, with a reduced computational burden one can easily include, in the problem formulation, a number of control specifications, such as the requirement to avoid critical regions in the state space or to reduce as much as possible the use of some actuators in other zones of the state space. The proposed MPC method has stability properties and is applied for control of a thermal system. The reported simulation results witness its favourable characteristics with respect to a standard MPC implementation.     

Output feedback MPC with guaranteed robust stability

This paper focuses on the issues of robust stability of model predictive control (MPC). The control problem is formulated as linear matrix inequalities (LMI) optimization problem. A suboptimal solution for the output feedback control problem is proposed. The size of the resulting MP controller is reduced by using a suitable state-space representation of the process. Guaranteed stability conditions for the output feedback MPC are enforced via a Lyapunov type constraint. An iterative algorithm is developed resulting in a pair of coupled LMI optimization problems which provide a robustly stable output feedback gain. Model uncertainties are considered via a polytopic set of process models. The methodology is illustrated with the simulation of the control problem of two chemical processes. The results show that the proposed strategy eliminates the need to detune the MP controller improving the performance for most of the cases considered.     

Enlarging the domain of attraction of stable MPC controllers, maintaining the output performance

This work presents an alternative way to formulate the stable Model Predictive Control (MPC) optimization problem that allows the enlargement of the domain of attraction, while preserving the controller performance. Based on the dual MPC that uses the null local controller, it proposed the inclusion of an appropriate set of slacked terminal constraints into the control problem. As a result, the domain of attraction is unlimited for the stable modes of the system, and the largest possible for the non-stable modes. Although this controller does not achieve local optimality, simulations show that the input and output performances may be comparable to the ones obtained with the dual MPC that uses the LQR as a local controller.     

Inversion-based optimal output tracking–transition switching with preview for nonminimum-phase linear systems

In this article, the problem of nonperiodic tracking–transition switching with preview is considered. Such a control problem exists in applications including nanoscale material property mapping, robot manipulation, and probe-based nanofabrication, where the output needs to track the desired trajectory during the tracking sections, and rapidly transit to another point during the transition sections with no post-transition oscillations. Due to the coupling between the control of the tracking sections and that of the transition ones, and the potential mismatch of the boundary system state at the tracking–transition switching instants, these control objectives become challenging for nonminimum-phase systems. In the proposed approach, the optimal desired output trajectory for the transition sections is designed through a direct minimization of the output energy, and the needed control input that maintains the smoothness of both the output and the system state across all tracking–transition switching is obtained through a preview-based stable-inversion approach. The needed preview time is quantified by the characteristics of the system dynamics, and can be minimized via the recently developed optimal preview-based inversion technique. The proposed approach is illustrated through a nanomanipulation example in simulation.     

INDIRECT ADAPTIVE CONTROL FOR NONLINEAR SYSTEMS WITH SYSTEM PERTURBATION

This paper presents an indirect adaptive control scheme for a class of input-output linearizable nonlinear systems subjected to system perturbations. System parameters are unknown and estimated recursively by a parameter estimator to obtain approximate system output and output derivatives, and then to derive an adaptive control law. In the parameter estimator, a dead-zone approach is used to avoid the parameter drift problem. A positive switching gain is also set to decrease the dead-zone value to obtain better output tracking performance. Under some assumptions, the indirect adaptive control scheme is proved to be stable.     

Stable MPC for tracking with maximal domain of attraction

In this work, a stable MPC that maximizes the domain of attraction of the closed-loop system is proposed. The proposed approach is suitable to real applications in the sense that it accounts for the case of output tracking, it is offset free if the output target is reachable and minimizes the offset if some of the constraints are active at steady state. The new approach is based on the definition of a Minkowski functional related to the input and terminal constraints of the stable infinite horizon MPC. It is also shown that the domain of attraction is defined by the system model and the constraints, and it does not depend on the controller tuning parameters. The proposed controller is illustrated with small order examples of the control literature.     

Hybrid static output feedback stabilization of two-dimensional LTI systems: a geometric method

For two-dimensional linear time-invariant (LTI) systems which are not stabilizable via a single static output feedback, we propose a hybrid stabilization strategy based on a geometric method. More precisely, we design two static output feedback gains and a switching law between the feedback gains so that the entire closed-loop system is asymptotically stable. The proposed switching law is composed of output-dependent switching and time-controlled switching. We demonstrate the hybrid control method with various examples for different cases.     

异步切换多智能体系统的协同输出调节

研究了一类控制器模态和系统模态不匹配的异步切换多智能体系统的输出调节问题.利用平均驻留时间和联合切换信号相结合的方法来处理由控制器与系统模态的切换存在时延引起的系统不稳定问题.提出一种基于输出反馈的切换控制策略,给出了异步切换多智能体输出调节问题可解的充分条件.最后,通过仿真实例验证结果的有效性.