Model Predictive control

Model Predictive Control (MPC) originated in the late seventies and has developed considerably since then. The term Model Predictive Control does not designate a specific control strategy but rather an ample range of control methods which make explicit use of a model of the process to obtain the control signal by minimizing an objective function. These design methods lead to controllers which have practically the same structure and present adequate degrees of freedom. The ideas, appearing in greater or lesser degree in the predictive control family, are basically:

This chapter describes the elements that are common to all Model Predictive controllers, showing the various alternatives used in the different implementations. Some of the most popular methods will later be reviewed to demonstrate their most outstanding characteristics.

As has been shown in previous chapters, there is a wide family of predictive controllers, each member of which is defined by the choice of the common elements such as the prediction model, the objective function and obtaining the control law.

This chapter describes one of the most popular predictive control algorithms: Generalized Predictive Control (GPC). The method is developed in detail, showing the general procedure to obtain the control law and its most outstanding characteristics. The original algorithm is extended to include the cases of measurable disturbances and change in the predictor. Close derivations of this controller such as CRHPC and Stable GPC are also treated here, illustrating the way they can be implemented.

One of the reasons for the success of the traditional PID controllers in industry is that PID are very easy to implement and tune using heuristic tuning rules such as the Ziegler-Nichols rules frequently used in practice. A Generalized Predictive Controller, as shown in the previous chapter, results in a linear control law which is very easy to implement once the controller parameters are known. The derivation of the GPC parameters requires, however, some mathematical complexities such as recursively solving the Diophantine equation, forming the matrices G, G′ and f and then solving a set of linear equations. Although this is not a problem for people in the research control community where mathematical packages are normally available, it may be discouraging for practitioners used to much simpler ways of implementing and tuning controllers.

Most industrial plants have many variables that have to be controlled (outputs) and many manipulated variables or variables used to control the plant (inputs). In certain cases a change in one of the manipulated variables mainly affects the corresponding controlled variable, and each the input-output pair can be considered as a single-input single-output (SISO) plant and controlled by independent loops. In many cases, when one of the manipulated variables is changed, it not only affects the corresponding controlled variable but also upsets the other controlled variables. These interactions between process variables may result in poor performance of the control process or even instability. When the interactions are not negligible, the plant must be considered to be a process with multiple inputs and outputs (MIMO) instead of a set of SISO processes. The control of MIMO processes has been extensively treated in literature; perhaps the most popular way of controlling MIMO processes is by designing decoupling compensators to suppress or diminish the interactions and then designing multiple SISO controllers. This requires first determining how to pair the input and output variables, that is, which manipulated variable will be used to control which output variables, and that the plant have the same number of manipulated and controlled variables. Total decoupling is very difficult to achieve for processes with complex dynamics or exhibiting dead times.

The control problem was formulated in the previous chapters considering all signals to possess an unlimited range. This is not very realistic because in practice all processes are subject to constraints. Actuators have a limited range of action and a limited slew rate, as is the case of control valves limited by a fully closed and fully open position and a maximum slew rate. Constructive or safety reasons, as well as sensor range, cause bounds in process variables, as in the case of levels in tanks, flows in pipes, and pressures in deposits. Furthermore, in practice, the operating points of plants are determined to satisfy economic goals and lie at the intersection of certain constraints. The control system normally operates close to the limits and constraint violations are likely to occur. The control system, especially for longrange predictive control, has to anticipate constraint violations and correct them in an appropriate way. Although input and output constraints are basically treated in the same way, as is shown in this chapter, the implications of the constraints differ. Output constraints are mainly due to safety reasons and must be controlled in advance because output variables are affected by process dynamics. Input (or manipulated) variables can always be kept in bound by the controller by clipping the control action to a value satisfying amplitude and slew rate constraints.

Mathematical models of real processes cannot contemplate every aspect of reality. Simplifying assumptions have to be made, especially when the models are going to be used for control purposes, where models with simple structures (linear in most cases) and sufficiently small size have to be used due to available control techniques and real-time considerations. Thus, mathematical models, especially control models, can only describe the dynamics of the process in an approximative way.

In general, industrial processes are nonlinear, but, as has been shown in this book, most MPC applications are based on the use of linear models. There are two main reasons for this: on one hand, the identification of a linear model based on process data is relatively easy and, on the other hand, linear models provide good results when the plant is operating in the neighbourhood of the operating point. In the process industries, where linear MPC is widespread, the objective is to keep the process around the stationary state rather than perform frequent changes from one operation point to another and, therefore, a precise linear model is enough. Besides, the use of a linear model together with a quadratic objective function gives rise to a convex problem (Quadratic Programming) whose solution is well studied with many commercial products available. The existence of algorithms that can guarantee a convergent solution in a time shorter than the sampling time is crucial in processes where a great number of variables appear.

In most processes there are not only continuous variables but also variables that have a discrete nature. For a long time, the control of processes with discrete variables and the control of processes with continuous variables were considered to be two completely different things. On the one hand, the theories of finite state machines were used to control processes with discrete variables, and on the other hand, linear and nonlinear control theory was used for the control of continuous variables. The techniques for modelling and analysis of these types of systems are different. In the case of continuous systems, differential equations, transfer functions, etc., are used as modelling tools, while in the discrete counterpart, state transition graphs, Petri Nets, etc., are employed (see ). From the beginning of the 1990s there has been great interest in processes that have both discrete and continuous parts. Hybrid systems are dynamic systems with both continuous-state and discrete-state and event variables. That is, the plant has time-driven and event-driven dynamics, the controller affects both time-driven and event-driven components, and it may deal with continuous and/or discrete signals.

One of the disadvantages of MPC is that the computation time required in some cases considerably limits the bandwidth of processes to which it can be applied. This is the case of MPC in the presence of constraints, adaptive MPC, robust MPC and MPC of nonlinear processes. This chapter is devoted to explaining some of the procedures used to reduce the amount of computation needed for the implementation of MPC. All of these procedures are based on doing most of the required computation off-line, leaving only part of the computation for the online part of the implementation.

This chapter is dedicated to presenting some MPC applications to the control of different real and simulated processes. The first application presented corresponds to a self-tuning and a gain scheduling GPC for a distributed collector field of a solar power plant. In order to illustrate how easily the control scheme shown in Chapter 5 can be used in any commercial control system, some applications concerning the control of typical variables such as flows, temperatures and levels of different processes of a pilot plant are presented. The description of two applications in the food industry (a sugar refinery and an olive oil mill) are included. Finally the application of an MPC to a highly nonlinear process (a mobile robot) is also described.