MPC for tracking zone regions

Abstract

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.

Introduction

In modern processing plants, MPC controllers are usually implemented as part of a multilevel hierarchy of control functions [1], [2]. At the intermediary levels of this control structure, the process unit optimizer computes an optimal economic steady state and passes this information to the MPC in a lower level for implementation. The role of the MPC is then to drive the plant to the most profitable operating condition, fulfilling the constraints and minimizing the dynamic error along the path. In many cases, however, the optimal economic steady state operating condition is not given by a point in the output space (fixed set-point), but is a region into which the output should lie most of the time. In general, based on operational requirements, process outputs can be classified into two broad categories: (1) set-point controlled, outputs to be controlled at a desired value, and (2) set-interval controlled, outputs to be controlled within a desired range. For instance, production rate and product quality may fall into the first category, whereas process variables, such as level, pressure, and temperature in different units/streams may fall into the second category. The reasons for using set-interval control in real applications may be several, and they are all related to the process degrees of freedom: (1) In some problems some inputs of a square system without degrees of freedom are desired to be steered to a specific steady state values (input set-points), and then to account for the lack of degrees of freedom, the use of output zone control arises naturally (for example, it could be desirable, by economic reasons, to drive feed-rate to its maximum). (2) In another class of problems, there are highly correlated outputs to be controlled, and there are not enough inputs to control them independently. Controlling the correlated outputs within zones or ranges is one solution for this kind of problem (for instance, controlling the dense and dilute phase temperatures on an FCC regenerator). (3) A third important class of zone control problems relates to using the surge capacity of tanks to smooth out the operation of a unit. In this case, it is desirable to let the level of the tank float between limits, as necessary, to buffer disturbances between sections of a plant. Conceptually, the output intervals are not output constraints, since they are steady state desired zones that can be transitorily disregarded, while the (dynamic) constraints must be respected at each time. In addition, the determination of the output intervals is related to the steady state operability of the process, and it is not a trivial problem. A special care should be taken about the compatibility between the available input set (given by the input constraints) and the desired output set (given by the output intervals). In [3], [4], for instance, an operability index that quantify how much of the region of the desired outputs can be achieved using the available inputs, taking into account the expected disturbance set, is defined. As a result a methodology to obtain the tightest possible operable set of achievable output steady state is derived. Then, the operating control intervals should be subsets of these tightest intervals. In practice, however, the operators are not usually aware of these maximum zones and may select control zones that are not fully consistent with the maximum zones and the operating control zones may be fully or partly unreachable. The MPC controller has to be robust to this poor selection of the control zones.

Model predictive control (MPC) is one of the most successful techniques of advanced control in the process industry. This is due to its control problem formulation, the natural usage of the model to predict the expected evolution of the plant, the optimal character of the solution and the explicit consideration of hard constraints in the optimization problem. Thanks to the recent developments of the underlying theoretical framework, MPC has become a mature control technique capable to provide controllers ensuring stability, robustness, constraint satisfaction and tractable computation for linear and for nonlinear systems [5]. The control law is calculated by predicting the evolution of the system and computing the admissible sequence of control inputs which makes the system evolves satisfying the constraints and minimizing the predicted cost. This problem can be posed as an optimization problem. To obtain a feedback policy, the obtained sequence of control inputs is applied in a receding horizon manner, solving the optimization problem at each sample time. Considering a suitable penalization of the terminal state and an additional terminal constraint, asymptotic stability and constraints satisfaction of the closed loop system can be proved [6].

Most of the MPC stability and feasibility results consider the regulation problem, that is steering the system to a fixed steady state (typically the origin). It is clear that for a given non-zero set-point, a suitable choice of the steady state can be chosen and the problem can be posed as a regulation problem translating the state and input of the system [7]. However, since the stabilizing choice of the terminal cost and constraints depends on the desired steady state, when the target operating point changes, the feasibility of the controller may be lost and the controller fails to track the reference [8], [9], [10], [11], thus requiring to re-design the MPC at each change of the reference. The computational burden that the design of a stabilizing MPC requires may make this approach not viable. For such case, the steady state target can be determined by solving an optimization problem that determines the steady state and input targets. This target calculation can be formulated as different mathematical programs for the cases of perfect target tracking or non-square systems [12], or by solving a unique problem for both situations [13]. In [14], an MPC for tracking is proposed, which is able to steer the system to any admissible set-point in an admissible way. The main characteristics of this approach are: an artificial steady state is considered as a decision variable, a cost that penalizes the error between the predicted variables and the artificial steady state is minimized, an additional term that penalizes the deviation between the artificial steady state and the steady state target is added to the cost function (the so-called offset cost function), and an invariant set for tracking is defined as extended terminal set. This controller ensures both, recursive feasibility and convergence to the target (if admissible) for any change of the steady state target. Furthermore, if the target is not admissible, the system is steered to the closest admissible steady state. In [15], the MPC for tracking is extended considering a general offset cost function. Under some mild sufficient assumptions, the new offset cost function ensures the local optimality property, letting the controller achieve optimal closed loop performance.

From the point of view of the controller, several approaches have been developed to account for the set-interval control. Ref. [16], which represents an excellent survey paper of the existing industrial MPC technology, describes a variety of industrial controller and mentions that they always provide a zone control option. That paper presents two ways to implement zone control: (1) defining upper and lower soft constraints, and (2) using the set-point approximation of soft constraints to implement the upper and lower zone boundaries (the DMC-plus algorithm). One of the main problems of these industrial controllers (as was stated in the same paper) is the lack of nominal stability. A second example of zone control can be found in [17], where the authors exemplify the application of this strategy to a FCC system. Although this strategy has shown to have an acceptable performance, stability cannot be proved, even if an infinite horizon is used, since the control system keeps switching from one controller to another throughout the continuous operation of the process. A third example is the closed loop stable MPC controller presented in [18]. In this approach, the authors develop a controller that considers the zone control of the system outputs and incorporates steady state economic targets in the control cost function. Assuming open-loop stable systems, classical stability proofs are extended to the zone control strategy by considering the output set-points as additional decision variables of the control problem. Furthermore, a set of slack variables is included into the formulation to assure both, recursive feasibility of the on-line optimization problem and convergence of the system inputs to the targets. This controller, however, is formulated for stable open-loop stable systems, and since it considers a null controller as local controller, it does not achieve local optimality. An extension of this strategy to the robust case, considering multi-model uncertainty, was proposed in [19].

From a theoretic point of view, the control objective of the zone control 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 and another. In this work, the controller proposed in [14], [15], is extended to deal with the zone control, generalizing the conditions of the offset cost function to use a distance to a convex target set. This controller ensures recursive feasibility and convergence to the target set for any stabilizable plant. This property holds for any class of convex target sets and also in the case of time-varying target sets. For the case of polyhedral target sets, several formulations of the controller are proposed that allows to derive the control law from the solution of a single quadratic programming problem. One of these formulations allows also to consider target points and target sets simultaneously in such a way that the controller steers the plant to the target point if reachable while it steers the plant to the target set in the other case. Finally, it is worth to remark that the proposed controller inherits the properties of the controller proposed in [14], [15].

This paper is organized as follows: in the following section the constrained tracking problem is stated. In Section 3 the new MPC for tracking is introduced and in Section 5 its implementation is presented. Finally an illustrative example is shown and some conclusions are drawn.

Notation and basic definitions: vector $(a\text{,}b)$ denotes $[a^T\text{,}{b}^T{]}^T$; for a given $\lambda$ and a given set X, $\lambda X\triangleq \{\lambda x:x\in X\}\text{;}\mathit{int}(X)$ denotes the interior of set X; a matrix T definite positive is denoted as $T>0$ and $T>P$ denotes that $T-P>0$. For a given symmetric matrix $P>0\text{,}\Vert x{\Vert }_P$ denotes the weighted euclidean norm of x, i.e. $\Vert x{\Vert }_P\triangleq \sqrt{x^T\mathit{Px}}$. Matrix ${\mathbf{0}}_{n\text{,}m}\in {\mathbb{R}}^{n\times m}$ denotes a matrix of zeros. Matrix ${\mathbf{I}}_m\in {\mathbb{R}}^{m\times m}$ denotes the identity matrix. Consider $a\in {\mathbb{R}}^{n_a}\text{,}b\in {\mathbb{R}}^{n_b}$, and set $\Gamma \subset {\mathbb{R}}^{n_a+n_b}$, then projection operation is defined as ${\mathit{Proj}}_a(\Gamma )\triangleq \{a\in {\mathbb{R}}^{n_a}:\exists b\in {\mathbb{R}}^{n_b}\text{,}(a\text{,}b)\in \Gamma \}$. Vector $\mathbf{u}(p)$ denotes the sequence of control action $\mathbf{u}(p)\triangleq \{u(0\text{;}p)\text{,}u(1\text{;}p)\text{,}\dots \}$, where p is a parameter. Given a subdifferential function $V_O(x)$, notation $\partial V_O(x)$ defines the subdifferential of $V_O(x)$ [20]. Set ${\mathrm{\Omega }}_{t\text{,}K}^w$ is the invariant set for tracking, considered in the augmented state $(x\text{,}\theta )$. $P_N^Z(x\text{,}{\Gamma }_t)$ is the optimization problem for the zone region tracking problem for an horizon of length N, in the set of parameters $(x\text{,}{\Gamma }_t)$. ${\Gamma }_t$ is the target zone.