Optimal MPC for tracking of constrained linear systems

Model predictive control (MPC) is one of the few techniques which is able to handle constraints on both state and input of the plant. The admissible evolution and asymptotic convergence of the closed-loop system is ensured by means of suitable choice of the terminal cost and terminal constraint. However, most of the existing results on MPC are designed for a regulation problem. If the desired steady-state changes, the MPC controller must be redesigned to guarantee the feasibility of the optimisation problem, the admissible evolution as well as the asymptotic stability. Recently, a novel MPC has been proposed to ensure the feasibility of the optimisation problem, constraints satisfaction and asymptotic evolution of the system to any admissible target steady-state. A drawback of this controller is the loss of a desirable property of the MPC controllers: the local optimality property. In this article, a novel formulation of the MPC for tracking is proposed aimed to recover the optimality property maintaining all the properties of the original formulation.     

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.     

MPC for tracking zone regions

This paper deals with the problem of tracking target sets using a model predictive control (MPC) law. Some MPC applications require a control strategy in which some system outputs are controlled within specified ranges or zones (zone control), while some other variables – possibly including input variables - are steered to fixed target or set-point. In real applications, this problem is often overcome by including and excluding an appropriate penalization for the output errors in the control cost function. In this way, throughout the continuous operation of the process, the control system keeps switching from one controller to another, and 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. 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. In this work, a stable MPC formulation for constrained linear systems, with several practical properties is developed for this scenario. The concept of distance from a point to a set is exploited to propose an additional cost term, which ensures both, recursive feasibility and local optimality. The performance of the proposed strategy is illustrated by simulation of an ill-conditioned distillation column.     

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.     

LMI-based robust model predictive control and its application to an industrial CSTR problem

In this paper, robust model predictive control (MPC) is studied for a class of uncertain linear systems with structured time-varying uncertainties. This general class of uncertain systems is useful for nonlinear plant modeling in many circumstances. The controller design is characterizing as an optimization problem of the “worst-case” objective function over infinite moving horizon, subject to input and output constraints. A sufficient state-feedback synthesis condition is provided in the form of linear matrix inequality (LMI) optimizations, and will be solved on-line. The stability of such a control scheme is determined by the feasibility of the optimization problem. To demonstrate its usefulness, this robust MPC technique is applied to an industrial continuous stirred tank reactor (CSTR) problem with explicit input and output constraints. Its relative merits to conventional MPC approaches are also discussed.     

Output feedback robust MPC with one free control move for the linear polytopic uncertain system with bounded disturbance

In this paper a previous approach, for the robust model predictive control (MPC) for a linear polytopic uncertain system, is extended to the case with bounded disturbance and unmeasurable state. The controller on-line optimizes a free control move followed by an output feedback control law based on the pre-specified state estimator. A key technique for this controller is an appropriate formulation of the estimation error bound which accounts for recursive feasibility of the optimization problem. The quadratic boundedness (QB) of the augmented state is guaranteed by the proposed approach. A numerical example is given to illustrate the effectiveness of the proposed controller.     

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.     

Robustness analysis of a hybrid ground coupled heat pump system with model predictive control

MPC of thermal systems usually results in robust operation with respect to uncertainties thanks to some key characteristics of the controller. However, the true limit until which these systems will actually be robust is rarely known explicitly. In this study a Hybrid Ground Coupled Heat Pump (HyGCHP) system with MPC is investigated, for which state estimation and disturbance prediction are highly uncertain, moreover, the system performance is highly sensitive to errors at these points. It has become popular to design control systems which perform explicit computations to assure robustness (e.g. min–max Robust MPC) but this framework is computationally demanding, therefore, not widely applied. An alternative is to perform robustness analysis of an MPC controlled system which is though generally avoided due to complicated theoretical formulations, implicitness and conservativeness of the approach. To tackle these issues an existing framework for robustness analysis is extended and applied to the case of a HyGCHP system with MPC to analyze robustness with respect to state estimation uncertainty. This paper presents an approach to use the original formulation, suggested for regulation/stabilization in order to analyze robustness for the case of set point tracking. The results show that the maximum allowed state estimation uncertainty found by robustness analysis of the regulation problem is confirmed by the simulated HyGCHP system with MPC, which performs set point tracking. In conclusion, the method gives a reliable guarantee for the degree of state estimation uncertainty, up to which the HyGCHP system investigated remains robust. Future research can extend the robustness analysis method towards disturbance prediction uncertainty.     

Model predictive control of an electric arc furnace off-gas process

This paper shows that an electric arc furnace off-gas system can provide valuable manipulated variables for feedback control which can improve furnace efficiency and contribute to safety in the workplace. Model predictive control (MPC) is used to illustrate this concept using practically motivated control objectives. An initial verification of a non-linear furnace model with plant data is shown. The design of MPC controllers for the furnace is discussed and results are shown by way of simulation. Evaluation of the final controller against traditional manual operation is done, and the setpoint tracking capability of the controller is tested.     

Methods for obtaining robust tracking control laws

This paper treats the problem of determining a tracking control law for linear systems with uncertain parameters. The tracking control is robust in the sense that it achieves asymptotic tracking of the command reference input for all admissible parameter values.Based on the available theoretical results for determining tracking controllers, numerical methods for obtaining the feedback gains for a robust control law are presented. The efficacy of these methods is demonstrated by applying them to two aircraft examples.     

Characteristic polynomial assignment for plants with semialgebraic uncertainty: A robust diophantine equation approach

In this paper, we address the problem of robust characteristic polynomial assignment for LTI systems whose parameters are assumed to belong to a semialgebraic uncertainty region. The objective is to design a dynamic fixed‐order controller in order to constrain the coefficients of the closed‐loop characteristic polynomial within prescribed intervals. First, necessary conditions on the plant parameters for the existence of a robust controller are reviewed, and it is shown that such conditions are satisfied if and only if a suitable Sylvester matrix is nonsingular for all possible values of the uncertain plant parameters. The problem of checking such a robust nonsingularity condition is formulated in terms of a nonconvex optimization problem. Then, the set of all feasible robust controllers is sought through the solution to a suitable robust diophantine equation. Convex relaxation techniques based on sum‐of‐square decomposition of positive polynomials are used to efficiently solve the formulated optimization problems by means of semidefinite programming. The presented approach provides a generalization of the results previously proposed in the literature on the problem of assigning the characteristic polynomial in the presence of plant parametric uncertainty. Copyright © 2014 John Wiley & Sons, Ltd.     

Robust MPC for Linear Systems with Structured Time‐Varying Uncertainties and Saturating Actuator

In this paper, a synthesis of model predictive control (MPC) algorithm is presented for uncertain systems subject to structured time‐varying uncertainties and actuator saturation. The system matrices are not exactly known, but are affine functions of a time varying parameter vector. To deal with the nonlinear actuator saturation, a saturated linear feedback control law is expressed into a convex hull of a group of auxiliary linear feedback laws. At each time instant, a state feedback law is designed to ensure the robust stability of the closed‐loop system. The robust MPC controller design problem is formulated into solving a minimization problem of a worst‐case performance index with respect to model uncertainties. The design of controller is then cast into solving a feasibility of linear matrix inequality (LMI) optimization problem. Then, the result is further extended to saturation dependent robust MPC approach by introducing additional variables. A saturation dependent quadratic function is used to reduce the conservatism of controller design. To show the effectiveness, the proposed robust MPC algorithms are applied to a continuous‐time stirred tank reactor (CSTR) process.     

Synthesis of Mixed Objective Output Feedback Robust Model Predictive Control

Aiming at the constrained polytopic uncertain system with energy‐bounded disturbance and unmeasurable states, a novel synthesis scheme to design the output feedback robust model predictive control(MPC)is put forward by using mixed H₂/H_∞ design approach. The proposed scheme involves an offline design of a robust state observer using linear matrix inequalities(LMIs)and an online output feedback robust MPC algorithm using the estimated states in which the desired mixed objective robust output feedback controllers are cast into efficiently tractable LMI‐based convex optimization problems. In addition, the closed‐loop stability and the recursive feasibility of the proposed robust MPC are guaranteed through an appropriate reformulation of the estimation error bound (EEB). A numerical example subject to input constraints illustrates the effectiveness of the proposed controller.     

基于容许集的多胞不确定系统三模RMPC

针对输入和状态受约束的多胞不确定线性系统,提出了基于容许集的扩大吸引域三模鲁棒模型预测控制方法.在多面体不变集离线模型预测控制算法的基础上引入容许集,以多面体不变集序列的并集作为模态1,基于N步容许集的控制容许集作为模态2,并利用离线设计和在线优化的控制策略,设计了三模变终端约束鲁棒模型预测控制算法,以实现系统渐近稳定.该算法不仅降低了在线运算量,而且扩大了吸引域.最后的仿真结果验证了所提出算法的有效性.     

移动机械手的跟踪控制

讨论一类不确定非完整移动机械手的跟踪控制问题。在系统惯性参数不精确知道的情况下,提出一种鲁棒控制器。为提高跟踪性能,进一步设计出使跟踪误差指数收敛的控制器。计算机仿真验证了所提出控制律的有效性。     

一类输入受限不确定时滞系统的准Min-Max模型预测控制

针对一类输入受限离散不确定时滞系统,提出一种基于准Ｍｉｎ-Ｍａｘ的模型预测控制器设计方法．定义了时滞系统的鲁棒性能指标,给出了系统稳定的充分条件,通过求解ＬＭＩ凸优化获得控制器．准Ｍｉｎ-Ｍａｘ预测控制将当前控制量作为独立优化变量,与其他作为反馈控制的时域控制序列分开处理,有效地降低了算法的保守性,提高了可行性．仿真算例验证了所提出控制方法的有效性．     

Guaranteed cost repetitive control for uncertain discrete-time systems

This paper considers the problem of guaranteed cost repetitive control for uncertain discrete-time systems. The uncertainty in the system is assumed to be norm-bounded and time-varying. The objective is to develop a novel design method for the repetitive control systems by introducing a robust output feedback controller so that the closed-loop system is quadratically stable and a certain bound of performance index is guaranteed for all admissible uncertainties. Necessary and sufficient conditions for the existence of such controllers are derived in terms of linear matrix inequality (LMI). A suboptimal guaranteed cost controller is obtained by solving an optimization problem with LMI constraints. Finally, a simulation example is provided to illustrate the validity of the proposed method.     

Robust distributed model predictive control of linear systems with structured time-varying uncertainties

In this work, synthesis of robust distributed model predictive control (MPC) is presented for a class of linear systems subject to structured time-varying uncertainties. By decomposing a global system into smaller dimensional subsystems, a set of distributed MPC controllers, instead of a centralised controller, are designed. To ensure the robust stability of the closed-loop system with respect to model uncertainties, distributed state feedback laws are obtained by solving a min–max optimisation problem. The design of robust distributed MPC is then transformed into solving a minimisation optimisation problem with linear matrix inequality constraints. An iterative online algorithm with adjustable maximum iteration is proposed to coordinate the distributed controllers to achieve a global performance. The simulation results show the effectiveness of the proposed robust distributed MPC algorithm.     

Synthesizing scheduled robust model predictive control with target tracking

Systems with large operating regions and non-zero state target tracking have limited the industrial application of robust model predictive control (RMPC) with synthetic action. To overcome the problem, this paper presents a novel formulation of synthesizing scheduled RMPC for linear time varying (LTV) systems. Off-line, we compute the matrix that transforms target output into steady state first. Then a set of stabilizing state feedback laws which are corresponding to a set of estimated regions of stability covering the desired operating region are provided. On-line, these control laws are implemented as a single scheduled state feedback model predictive control (MPC) which switches between the set of local controllers and achieve the desired target at last. Finally, the algorithm is illustrated with an example.     

Hybrid Model Predictive Control of Exhaust Recompression Hcci

Homogeneous charge compression ignition (HCCI) holds promise for reduced emissions and increased efficiency compared to conventional internal combustion engines. As HCCI lacks direct actuation over the combustion phasing, much work has been devoted to designing controllers capable of set‐point tracking and disturbance rejection. This paper presents results on model predictive control (MPC) of the combustion phasing in an HCCI engine based on a hybrid model formulation composed of several linearizations of a physics‐based nonlinear model. The explicit representation of the MPC was implemented experimentally and the performance during set point changes was compared to that of a switched state feedback controller. The hybrid MPC produced smoother transients without overshoot when the set point change traversed several linearizations.