Model predictive control constraints

Introduction The model-based contfol strategy that has been most widely applied in the process industries is model predictive control (MFC). It is a general method that is especially well-suited for difficult multiinput, multioutput (MIMO) control problems where there are significant interactions between the manipulated inputs and the controlled outputs. Unlike other model-based control strategies, MFC can easily accommodate inequahty constraints on input and output variables such as upper and lower limits or rate-of-change limits. 

Comparison of the system behavior using three different model predictive controllers (a) minimum variance, (b) input constraint, (c) input penalty. 

Patwardhan, A. A., Rawlings, J. B., and Edgar, T. F., Model predictive control of nonlinear processes in the presence of constraints, presented at annual AIChE Meeting, Washington, D.C. (1988). 

The minimization of the quadratic performance index in Eq. (8-64), subject to the constraints in Eqs. (8-67) to (8-69) and the step response model in Eq. (8-61), can be formulated as a standard QP (quadratic programming) problem. Consequently, efficient QP solution techniques can be employed. When the inequality constraints in Eqs. (8-67) to (8-69) are omitted, the optimization problem has an analytical solution (Camacho and Bordons, Model Predictive Control, 2d ed., Springer-Verlag, New York, 2004 Maciejowski, Predictive Control with Constraints, Prentice-Hall, Upper Saddle River, N.J., 2002). If the quadratic terms in Eq. (8-64) are replaced by linear terms, an LP (linear programming) problem results that can also be solved by using standard methods. This MPC formulation for SISO control problems can easily be extended to MIMO problems. 

One important class of nonlinear programming techniques is called quadratic programming (QP), where the objective function is quadratic and the constraints are linear. While the solution is iterative, it can be obtained quickly as in linear programming. This is the basis for the newest type of constrained multivariable control algorithms called model predictive control, which is heavily used in the refining industry. See the earlier subsection on model predictive control for more details. 

Model predictive control (MPC) was developed in the 1970s and 1980s to meet control challenges of refineries. The advantages of MPC are most evident when it is used as a multivariable controller integrated with an optimizer. The greatest MPC benefits are realized in applications with dead-time dominance, interactions, constraints, and the need for optimization. As opposed to a traditional control loop, where the controller responds to a difference (error) between the set point and measurement, the predictive controller uses a vector difference between the future trajectory of the set point and the predicted trajectory of the controlled variable as its input (Figure 2.52). 

Model predictive control was conceived for multivariable systems with changing objectives and constraints. In simpler situations, a PID controller tuned according to internal model control (IMC) principles [8] can deliver equal performance with much less effort. 

Model predictive control is particularly useful when several control valves (or manipulators) affect an output of interest (what is called interaction) and also when some sort of constraint comes into play either on the inputs or on some measured variable. Since the controller itself knows about these interactions and constraints, it can in theory avoid those perils. It is important to remember that MPC merely suggests that the controller can predict the process response into the future, only to be checked (and corrected) by the next round of measurements. 

Very few unbiased publications have appeared in the literature comparing control effectiveness using MPC versus a well-designed conventional control system. Most of the MPC applications reported have considered fairly simple processes with a small number of manipulated variables. There are no published reports that discuss the application of MPC to an entire complex chemical plant, with one notable exception. That is the work of Ricker (1996), who compared MPC with conventional PI control for the Eastman process (TE problem). His conclusion was there appears to be little, if any, advantage to the use of nonlinear model predictive control (NMPC) in this application. In particular, the decentralized strategy does a better job of handling constraints—an area in which NMPC is reputed to excel.51... 

Schwarm, A., and Nikolaou, M., Chance constraint formulation of model predictive control, AIChE Annual Meeting (1997). 

Zafiriou, E., On the effect of tuning parameters and constraints on the robustness of model predictive controllers, Proceedings of Chemical Process Control—CPC TV, 363-393 (1991). 

The challenges in SMB control are not only the complex nonlinear process dynamics, but also the long delays of the effect of disturbances. The required control strategy has to be able to handle multivariable dynamics with time-delays and hard constraints. Model Predictive Control (MPC) has been proven to be the most effective control strategy for this type of problems [1,2]. Only recently, a few scientific publications have addressed the automatic... 

A linear model predictive control law is retained in both cases because of its attracting characteristics such as its multivariable aspects and the possibility of taking into account hard constraints on inputs and inputs variations as well as soft constraints on outputs (constraint violation is authorized during a short period of time). To practise model predictive control, first a linear model of the process must be obtained off-line before applying the optimization strategy to calculate on-line the manipulated inputs. The model of the SMB is described in [8] with its parameters. It is based on the partial differential equation for the mass balance and a mass transfer equation between the liquid and the solid phase, plus an equilibrium law. The PDE equation is discretized as an equivalent system of mixers in series. A typical SMB is divided in four zones, each zone includes two columns and each column is composed of twenty mixers. A nonlinear Langmuir isotherm describes the binary equilibrium for each component between the adsorbent and the liquid phase. 

The model predictive control used includes all features of Quadratic Dynamic Matrix Control [19], furthermore it is able to take into account soft output constraints as a non linear optimization. The programs are written in C++ with Fortran libraries. The manipulated inputs (shown in cm Vs) calculated by predictive control are imposed to the full nonlinear model of the SMB. The control simulations were made to study the tracking of both purities and the influence of disturbances of feed flow rate or feed composition. Only partial results are shown. 

The constraint-control problem is a multivariable square control problem, and various controllers can be used, such as a discrete integral controller or a model predictive controller. 

Model Predictive Control, also referred as moving or receding horizon control, has become an attractive control strategy especially for linear but also for nonlinear systems subject to input, state or output constraints [4]. MPC determines the control action based... 

At the heart of an model predictive control (MPC) application is the optimization of a variable subject to constraints. A typical MPC cost functional is given as follows ... 

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