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.     

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.     

Modified reference tracking MPC for constrained linear systems

This article presents a model predictive control for tracking piecewise constant references with a new steady-state parametrisation. The modified algorithm is based on the artificial reference idea, but the number of decision variables is equal to the standard MPC for regulation. The proposed strategy is able to track admissible constant references with an admissible evolution. If the reference is not admissible, the system is steered to the closest admissible stationary point. A modified initialisation algorithm is proposed to recover the enlarged domain of attraction provided by related artificial reference-based strategies. Simulation examples are presented to illustrate the benefits of the proposed strategy.     

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.     

Parsimonious cooperative distributed MPC algorithms for offset-free tracking

We propose in this paper novel cooperative distributed MPC algorithms for tracking of piecewise constant setpoints in linear discrete-time systems. The available literature for cooperative tracking requires that each local controller uses the centralized state dynamics while optimizing over its local input sequence. Furthermore, each local controller must consider a centralized target model. The proposed algorithms instead use a suitably augmented local system, which in general has lower dimension compared to the centralized system. The same parsimonious parameterization is exploited to define a target model in which only a subset of the overall steady-state input is the decision variable. Consequently the optimization problems to be solved by each local controller are made simpler. We also present a distributed offset-free MPC algorithm for tracking in the presence of modeling errors and disturbances, and we illustrate the main features and advantages of the proposed methods by means of a multiple evaporator process case study.     

A tracking MPC formulation that is locally equivalent to economic MPC

The stability proof for economic Model Predictive Control (MPC) is in general difficult to establish. In contrast, tracking MPC has well-established and practically applicable stability guarantees, but can yield poor closed-loop performance in terms of the selected economic criterion. In this paper, we derive a formal procedure to design a tracking MPC scheme so as to locally approximate the behaviour of economic MPC. Given an economic stage cost, the desired tracking stage cost can therefore be computed automatically. Because tracking MPC guarantees stability of the closed-loop system, our procedure succeeds if and only if economic MPC is locally stabilising. This fact can be used to certify whether economic MPC is not stabilising. We illustrate the theoretical developments in a simulated example.     

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.     

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.     

Latent variable MPC for trajectory tracking in batch processes

A novel multivariate empirical model predictive control strategy (LV-MPC) for trajectory tracking and disturbance rejection for batch processes is presented. The strategy is based on dynamic principal component analysis (PCA) models of the batch process. The solution to the control problem is computed in the low dimensional latent variable space of the PCA model. The trajectories of all variables over the future horizon are then computed from the latent variable solution of the controller. The excellent control performance and the modest closed-loop data requirements for identification are illustrated for the temperature tracking in simulations of an emulsion polymerization process, an exothermic chemical reaction system and for MIMO temperature and pressure tracking in a nylon polymerization autoclave.     

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 enhancement for tracking of complex profiles — The basic technique

This paper develops a unique approach that can be imposed on conventional predictive control schemes to provide tighter control when tracking complex setpoint trajectories. The method introduces a correction parameter η, independent of plant gain, evaluated online at each control timestep to drive the plant output to its reference more accurately than the original schemes. The strategy was applied to various systems that possess deadtime, higher order and multivariable characteristics with improved closed loop control. This is evidenced by significant reductions of mean square errors when using η. The strategy was applied on a nonlinear system and a practical setup, with reduced tracking error.     

On the solution of the tracking problem for non-linear systems with MPC

This paper presents a Model Predictive Control (MPC) algorithm for non-linear systems which solves the tracking problem for asymptotically constant references. Closed-loop stability of the equilibrium and asymptotic zero-error regulation are guaranteed. The performance of the method is discussed with the classical Continuous Stirred Tank Reactor (CSTR) control application.     

A RBF-ARX model-based robust MPC for tracking control without steady state knowledge

A RBF-ARX modeling and robust model predictive control (MPC) approach to achieving output-tracking control of the nonlinear system with unknown steady-state knowledge is proposed. On the basis of the RBF-ARX model with considering the system time delay, a local linearization state-space model is obtained to represent the current behavior of the nonlinear system, and a polytopic uncertain linear parameter varying (LPV) state-space model is built to represent the future system’s nonlinear behavior. Based on the two models, a quasi-min–max MPC algorithm with constraint is designed for output-tracking control of the nonlinear system with unknown steady state knowledge. The optimization problem of the quasi-min–max MPC algorithm is finally converted to the convex linear matrix inequalities (LMIs) optimization problem. Closed-loop stability of the MPC strategy is guaranteed by the use of parameter-dependent Lyapunov function and feasibility of the LMIs. Two examples, i.e. the modeling and control of a continuously stirred tank reactor (CSTR) and a two tank system demonstrate the effectiveness of the RBF-ARX modeling and robust MPC approach.     

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.     

Constrained nonlinear MPC for accelerated tracking piece-wise references and its applications to thermal systems

In this paper, we propose a model predictive control (MPC) strategy for accelerated offset-free tracking piece-wise constant reference signals of nonlinear systems subject to state and control constraints. Some special contractive constraints on tracking errors and terminal constraints are embedded into the tracking nonlinear MPC formulation. Then, recursive feasibility and closed-loop convergence of the tracking MPC are guaranteed in the presence of piece-wise references and constraints by deriving some sufficient conditions. Moreover, the local optimality of the tracking MPC is achieved for unreachable output reference signals. By comparing to traditional tracking MPC, the simulation experiment of a thermal system is used to demonstrate the acceleration ability and the effectiveness of the tracking MPC scheme proposed here.     

Closed-loop model re-identification of processes under MPC with zone control

This work deals with a procedure for model re-identification of a process in closed loop with an already existing commercial MPC. The controller considered here has a two-layer structure where the upper layer performs a target calculation based on a simplified steady-state optimization of the process. Here, it is proposed a methodology where a test signal is introduced in a tuning parameter of the target calculation layer. When the outputs are controlled by zones instead of at fixed set points, the approach allows the continuous operation of the process without an excessive disruption of the operating objectives as process constraints and product specifications remain satisfied during the identification test. The application of the method is illustrated through the simulation of two processes of the oil refining industry.     

Indefinite linear MPC and approximated economic MPC for nonlinear systems

The stability proof for economic model predictive control (MPC) relies on strict dissipativity, which is in general hard to check. In this contribution, we will first analyze the linear quadratic case with possibly indefinite cost. The conditions for stability will be recalled and it will be shown that every stabilizing LQR/MPC has an equivalent positive definite LQR/MPC which yields the same open loop and closed loop behavior. This analysis will then be used to formulate an approximated nonlinear economic MPC scheme, which has stability guarantees. An example will be used to illustrate the proposed technique and show its potential in terms of performance.     

Robust nonlinear MPC without terminal constraint for tracking changing setpoints in the presence of non-additive unknown disturbance

This paper develops a novel robust tracking model predictive control (MPC) without terminal constraint for discrete-time nonlinear systems capable to deal with changing setpoints and unknown non-additive bounded disturbances. The MPC scheme without terminal constraint avoids difficult computations for the terminal region and is thus simpler to design and implement. However, the existence of disturbances and/or sudden changes in a setpoint may lead to feasibility and stability issues in this method. In contrast to previous works that considered changing setpoints and/or additive slowly varying disturbance, the proposed method is able to deal with changing setpoints and non-additive non-slowly varying disturbance. The key idea is the addition of tightened input and state (tracking error) constraints as new constraints to the tracking MPC scheme without terminal constraints based on artificial references. In the proposed method, the optimal tracking error converges asymptotically to the invariant set for tracking, and the perturbed system tracking error remains in a variable size tube around the optimal tracking error. Closed-loop input-to-state stability and recursive feasibility of the optimization problem for any piece-wise constant setpoint and non-additive disturbance are guaranteed by tightening input and state constraints as well as weighting the terminal cost function by an appropriate stabilizing weighting factor. The simulation results of the satellite attitude control system are provided to demonstrate the efficiency of the proposed predictive controller.     

基于平面区域跟踪的目标位姿参数自动测量

提出一种从序列图像中自动跟踪测量目标位置和姿态参数的方法。利用单应性原理和上一帧图像中目标位姿参数的测量结果,将目标上的典型平面区域重建为同时含有几何信息和亮度信息的平面区域模板;然后根据投影方程,将该模板在一定的位置姿态参数下进行投影仿真成像,当模板的仿真成像结果与当前帧图像中的该平面区域达到最佳匹配时,认为此时仿真成像的位置姿态参数即为当前帧图像的测量结果。通过对该匹配问题进行最优化建模和求解,实现了序列图像中目标位姿参数的自动测量。实验结果表明,本文方法能够在序列图像中对含有典型平面区域的目标实现较高精度的自动跟踪测量。     

MPC for continuous piecewise-affine systems

A large class of hybrid systems can be described by a max–min-plus-scaling (MMPS) model (i.e., using the operations maximization, minimization, addition and scalar multiplication). First, we show that continuous piecewise-affine systems are equivalent to MMPS systems. Next, we consider model predictive control (MPC) for these systems. In general, this leads to nonlinear, nonconvex optimization problems. We present a new MPC method for MMPS systems that is based on canonical forms for MMPS functions. In case the MPC constraints are linear constraints in the inputs only, this results in a sequence of linear optimization problems such that the MPC control can often be computed in a much more efficient way than by just applying nonlinear optimization as was done in previous work.