**1. Introduction**

46 Frontiers of Model Predictive Control

Isidori A. (1985). Lecture Notes in Control and Information Sciences, Springer-Verlag, ISBN

Margellos, K.; Lygeros, J. (2010), *Proceedings of 49th IEEE Conference on Decision and Control*,

Poulsen, N. K.; Kouvaritakis, B.; Cannon, M. (2001a). Constrained predictive control and its

Poulsen, N. K.; Kouvaritakis, B.; Cannon, M. (2001b). Nonlinear constrained predictive

Slotine, J. E. ;Li W. (1991). Applied Nonlinear Control, Prentice-Hall, ISBN 0-13-040049-1,

Zietkiewicz, J. (2010), Nonlinear constrained predictive control of exothermic reactor,

*Robotics*, ISBN 978-989-8425-02-7, Vol.3, pp.208-212, Funchal, Portugal

application to a coupled-tanks apparatus, International Journal of Control, pp. 74:6,

control applied to a coupled-tanks apparatus, IEE Proc. Of Control Theory and

*Proceedings of 7th International Conference on Informatics in Control, Automation and* 

3-540-15595-3, ISBN 0-387-15595-3, Berlin, Germany

ISBN 978-1-4244-7745-6, Atlanta, GA

Applications, pp.17-24, ISNN 1350-2379

552-564, ISSN 1366-5820

New Jersey, USA

### **1.1 Aim of the chapter**

Predictive Control optimization problems may be rendered infeasible in the presence of constraints due to model-plant mismatches, external perturbations, noise or faults. This may cause the optimizer to issue a control sequence which is impossible to implement, leading to prediction errors, as well as loss of stability of the control loop. Such a problem motivates the development of techniques aimed at recovering feasibility without violating hard physical constraints imposed by the nature of the plant. Currently, setpoint management approaches and techniques dealing with changes in the constraints are two of the most effective solutions to recover feasibility with low computational demand. In this chapter a review of techniques that can be understood as one of the aforementioned is presented along with some illustrative simulation examples.

### **1.2 Concepts and literature review**

One of the main advantages of Predictive Control is the ability to deal with constraints over the inputs and states of the plant in an explicit manner, which brings better performance and more safety to the operation of the plant (Maciejowski, 2002), (Rossiter, 2003). Constraints over the excursion of the control signals are particularly common in processes that operate near optimal conditions (Rodrigues & Odloak, 2005). However, if the optimization becomes infeasible, possibly due to model-plant mismatches, external perturbations, noise or faults, a control sequence which is impossible to implement may be issued, leading to prediction errors, as well as loss of stability of the control loop (Maciejowski, 2002). Such a problem motivates the development of techniques aimed at recovering feasibility without violating hard physical constraints imposed by the nature of the plant.

The MPC formulation itself allows for a simple solution, which consists of enlarging the horizons, as means to allow for more degrees of freedom in the optimization. On the other hand, an increase in the computational burden associated to the solution of the optimization problem results, since there are more decision variables as well as constraints. Moreover, enlarging the horizons cannot solve all sorts of infeasibilities.

Constraint relaxation is one alternative which involves less decision variables and is usually effective. Nevertheless, it is often not obvious which constraints to relax and the amount by which they should be relaxed in order to attain a feasible optimization problem. There are

artificially reduce the distance between the actual plant state and the constraint set. The reference governor proposed by Kapasouris et al. (1988) inspired many techniques to deal with problems involving actuator saturation through manipulation of the setpoint or the tracking error (Gilbert & Kolmanovsky, 1995). There are also papers aiming at imposing a reference model to the behavior of the plant that employ setpoint management in order to

Infeasibility Handling in Constrained MPC 49

Stability guarantees may be achieved with setpoint management by using a terminal constraint invariant set parameterized by the setpoint. Limon et al. (2008) employ this technique parameterizing the terminal set in terms of the control and state setpoints. The authors show that an optimal management of the setpoint may be achieved, guaranteeing the smallest distance between the desired setpoint and the one used by the MPC. This procedure

An application of the parameterization of the terminal set in terms of the steady-state value of the control can be found in Almeida & Leissling (2010). In that paper, the technique is employed to circumvent infeasibility caused by actuator faults which limit the range of values of control that the actuator can deploy. On the other hand, in Afonso & Galvão (2010b) the authors manage the setpoint of a state variable that does not affect the control setpoint, making parameterization of the terminal set unnecessary, as means to overcome infeasibility brought

In this chapter, the treatment of infeasibility in the optimization problem of constrained MPC will be discussed. Some illustrative simulations will provide a basic coverage of this topic, which is of great importance to practical implementations of MPC due to the capability of circumventing problems brought about by model-plant mismatch, faults, noise, disturbances or simply reducing the computational burden required to calculate an adequate control

*v*ˆ \*(*k* | *k*)

+

*ref u*

*u* (*k* | *k*) *<sup>p</sup>*

−

*K*

+

**Plant**

*x*(*k*)

*ref x*

+

−

*x* (*k*) *<sup>P</sup>*

obtain feasibility when the control signals are bounded (Montandon et al., 2008).

increases the domain of attraction of the controller dramatically.

about by similar actuator faults.

**2. Adopted MPC formulation**

**Cost**

*i , , M v k i k* 1 K ˆ( 1| )

+ −

=

**Optimizer**

**Function Constraints**

**Prediction Model**

*i N x k i k* 1, , ˆ( | ) = K +

**Predictive Controller**

Fig. 1. MPC with inner feedback loop.

sequence.

different approaches for this purpose, some of which will be briefly discussed in this chapter. Initially, one must differentiate between two types of constraints (Alvarez & de Prada, 1997), (Vada et al., 2001):

**Physical constraints**: those limits that can never be surpassed and are determined by the physical functioning of the system. For instance, a valve cannot be opened more than 100% or less than 0%.

**Operating constraints**: those limits fixed by the plant operator. These limits, which are usually more restrictive than the physical constraints, define the band within which the variables are expected to be under normal operating conditions. For instance, it may be more profitable to operate a chemical reactor in a certain range of temperatures, in order to favor the kinetics of the desired reaction that forms products of economical interest. However, if maintaining such operating condition would compromise the safety of operation of the plant at some point, then the associated constraints could be relaxed.

The literature has many different approaches to constraint relaxation. Some infeasibility handling techniques are described in Rawlings & Muske (1993) and Scokaert & Rawlings (1999):

**Minimal time approach**: An algorithm identifies the smallest time, *κ*(*x*), which depends on the current state *x*, beyond which the state constraint can be satisfied over an infinite horizon. Prior to time *κ*(*x*), the state constraint is ignored, and the control law enforces the state constraint only after that time. An advantage of this method is that it leads to the earliest possible constraint satisfaction. Transient constraint violations, however, can be large.

**Soft-constraint approach**: Violations of the state constraints are allowed, but an additional term is introduced in the cost function to penalize the constraint violation.

In Zafiriou & Chiou (1993) the authors propose a method for calculating the smallest magnitude of the relaxation that renders the optimization feasible for a SISO system.

The paper by Scokaert (1994) presents many suggestions to circumvent the problem of infeasibility, among which, one that classifies the constraints in priority levels and tries to enforce the ones with higher priority through relaxation of the others.

Scokaert & Rawlings (1999) introduce an approach capable of minimizing the peak and duration of the constraint violation, with advantages concerning the transient response.

A relaxation procedure that can be applied either to the controls or to the system outputs is described by Alvarez & de Prada (1997). The control-related approach consists of relaxing the operating constraints on the control amplitude or rate of change according to a priority schedule. The output-related approach consists of relaxing the operating constraints on the output amplitude or modifying the time interval where such constraints are imposed within the prediction horizon.

In Vada et al. (2001) the proposed scheme involves the classification of the constraints in priority levels and the solution of a linear programming problem parallel to the MPC optimization. In Afonso & Galvão (2010a), different weights are employed for the relaxation of operating output constraints, up to the values of physical constraints, as means to overcome infeasibility caused by actuator faults.

Another alternative to recover feasibility are the so-called setpoint management procedures (Bemporad & Mosca, 1994), (Gilbert & Kolmanovsky, 1995), (Bemporad et al., 1997), which 2 Will-be-set-by-IN-TECH

different approaches for this purpose, some of which will be briefly discussed in this chapter. Initially, one must differentiate between two types of constraints (Alvarez & de Prada, 1997),

**Physical constraints**: those limits that can never be surpassed and are determined by the physical functioning of the system. For instance, a valve cannot be opened more than 100% or

**Operating constraints**: those limits fixed by the plant operator. These limits, which are usually more restrictive than the physical constraints, define the band within which the variables are expected to be under normal operating conditions. For instance, it may be more profitable to operate a chemical reactor in a certain range of temperatures, in order to favor the kinetics of the desired reaction that forms products of economical interest. However, if maintaining such operating condition would compromise the safety of operation of the plant

The literature has many different approaches to constraint relaxation. Some infeasibility handling techniques are described in Rawlings & Muske (1993) and Scokaert & Rawlings

**Minimal time approach**: An algorithm identifies the smallest time, *κ*(*x*), which depends on the current state *x*, beyond which the state constraint can be satisfied over an infinite horizon. Prior to time *κ*(*x*), the state constraint is ignored, and the control law enforces the state constraint only after that time. An advantage of this method is that it leads to the earliest possible constraint satisfaction. Transient constraint violations, however, can be large.

**Soft-constraint approach**: Violations of the state constraints are allowed, but an additional

In Zafiriou & Chiou (1993) the authors propose a method for calculating the smallest

The paper by Scokaert (1994) presents many suggestions to circumvent the problem of infeasibility, among which, one that classifies the constraints in priority levels and tries to

Scokaert & Rawlings (1999) introduce an approach capable of minimizing the peak and duration of the constraint violation, with advantages concerning the transient response.

A relaxation procedure that can be applied either to the controls or to the system outputs is described by Alvarez & de Prada (1997). The control-related approach consists of relaxing the operating constraints on the control amplitude or rate of change according to a priority schedule. The output-related approach consists of relaxing the operating constraints on the output amplitude or modifying the time interval where such constraints are imposed within

In Vada et al. (2001) the proposed scheme involves the classification of the constraints in priority levels and the solution of a linear programming problem parallel to the MPC optimization. In Afonso & Galvão (2010a), different weights are employed for the relaxation of operating output constraints, up to the values of physical constraints, as means to overcome

Another alternative to recover feasibility are the so-called setpoint management procedures (Bemporad & Mosca, 1994), (Gilbert & Kolmanovsky, 1995), (Bemporad et al., 1997), which

magnitude of the relaxation that renders the optimization feasible for a SISO system.

term is introduced in the cost function to penalize the constraint violation.

enforce the ones with higher priority through relaxation of the others.

at some point, then the associated constraints could be relaxed.

(Vada et al., 2001):

less than 0%.

(1999):

the prediction horizon.

infeasibility caused by actuator faults.

artificially reduce the distance between the actual plant state and the constraint set. The reference governor proposed by Kapasouris et al. (1988) inspired many techniques to deal with problems involving actuator saturation through manipulation of the setpoint or the tracking error (Gilbert & Kolmanovsky, 1995). There are also papers aiming at imposing a reference model to the behavior of the plant that employ setpoint management in order to obtain feasibility when the control signals are bounded (Montandon et al., 2008).

Stability guarantees may be achieved with setpoint management by using a terminal constraint invariant set parameterized by the setpoint. Limon et al. (2008) employ this technique parameterizing the terminal set in terms of the control and state setpoints. The authors show that an optimal management of the setpoint may be achieved, guaranteeing the smallest distance between the desired setpoint and the one used by the MPC. This procedure increases the domain of attraction of the controller dramatically.

An application of the parameterization of the terminal set in terms of the steady-state value of the control can be found in Almeida & Leissling (2010). In that paper, the technique is employed to circumvent infeasibility caused by actuator faults which limit the range of values of control that the actuator can deploy. On the other hand, in Afonso & Galvão (2010b) the authors manage the setpoint of a state variable that does not affect the control setpoint, making parameterization of the terminal set unnecessary, as means to overcome infeasibility brought about by similar actuator faults.

In this chapter, the treatment of infeasibility in the optimization problem of constrained MPC will be discussed. Some illustrative simulations will provide a basic coverage of this topic, which is of great importance to practical implementations of MPC due to the capability of circumventing problems brought about by model-plant mismatch, faults, noise, disturbances or simply reducing the computational burden required to calculate an adequate control sequence.
