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One of the main advantages of predictive control approaches is the capability of dealing explicitly with constraints on the manipulated and output variables. However, if the predictive control formulation does not consider model uncertainties, then the constraint satisfaction may be compromised. A solution for this inconvenience is to use robust model predictive control (RMPC) strategies based on linear matrix inequalities (LMIs). However, LMI-based RMPC formulations typically consider only symmetric constraints. This paper proposes a method based on pseudoreferences to treat asymmetric output constraints in integrating SISO systems. Such technique guarantees robust constraint satisfaction and convergence of the state to the desired equilibrium point. A case study using numerical simulation indicates that satisfactory results can be achieved.

Model-based predictive control (MPC) is a strategy in which a sequence of control actions is obtained by minimizing a cost function considering the predictions of a process model within a certain prediction horizon. At each sample time, only the first value of this sequence is applied to the plant, and the optimization is repeated in order to use feedback information [

In this context, Kothare et al. [

Within this scope, Cavalca et al. [

The remainder of this paper is organized as follows. Section

Throughout the text,

Consider a linear time-invariant system described by an uncertain model of the following form:

At each time

This min-max problem can be replaced with the following convex optimization problem with variables

If the problem (

Symmetric constraints on the manipulated variables of the form

Let

If

The proof of Lemma

As shown in Appendix A of Kothare et al. [

For a regulation problem around a point different from the origin, a change of variables can be used, so that the new origin corresponds to the desired equilibrium point [

The RMPC problem formulation presented in this section considers that the matrices

The present work is concerned with regulation problems around the origin involving a SISO system with output variable

However, this procedure may not be convenient in the following cases:

In these cases, the initial value of the output is admissible under the original asymmetric constraints, but not under the more conservative constraint (

(a) Illustration of the more conservative constraint (

Cavalca et al. [

Unlike the approach described above, which involves a pseudoreference

Illustration of the proposed pseudoreference scheme.

Given an initial state

It is assumed that the set

Define the pseudoreferences

For each

Let

Solve the problem

The matrices

Result of the PR algorithm.

The PR algorithm is said to be feasible if (

Let

Read the state

If

Let

Let

Calculate

Apply

Let

The main result of this work is stated in the following theorem, which is concerned with the satisfaction of constraints and convergence of the state trajectory to the origin under the control law given by the CPR algorithm.

If the PR algorithm is feasible and the CPR algorithm is applied to control the plant, then

For

A discrete state-space model of a double integrator will be employed to illustrate the proposed strategy. The matrices of the model are given by

The initial condition is set to

In this case, the set

In fact, the state variables

The pseudoreferences

Pseudoreferences and associated output constraints.

0 | −4.95 | 5.05 |

1 | −2.425 | 2.525 |

2 | −1.1625 | 1.2625 |

3 | −0.53125 | 0.63125 |

4 | −0.21563 | 0.31562 |

5 | −0.057813 | 0.15781 |

6 | 0 | 0.1 |

As discussed in Section

On the other hand, if the constraints are relaxed by setting

Simulation results using relaxed output constraints.

These findings motivate the adoption of the proposed strategy for handling the asymmetric output constraints. Figures

Simulation results using the proposed strategy: (a) output and (b) control signals.

The commutation between the successive pseudoreferences is illustrated in Figure

Commutation between pseudoreferences.

This paper presented a strategy for handling asymmetric output constraints within the scope of an LMI-based RMPC scheme. For this purpose, a procedure for defining a sequence of pseudoreferences was devised, along with a rule for commutation from one pseudoreference to the next. The proposed approach guarantees constraint satisfaction and convergence of the state trajectory to the origin, provided that the algorithm for determination of the pseudoreferences is feasible. The results of a numerical simulation study indicated that the proposed procedure may be a suitable alternative to the use of either more conservative constraints (which may lead to infeasibility issues) or more relaxed constraints (which do not guarantee satisfaction of the original restrictions). Future research could be concerned with the extension of the proposed approach to multiple input-multiple output (MIMO) systems. In this case, it may be necessary to define different pseudoreferences for each constrained output under consideration.

The authors gratefully acknowledge the support of FAPESP (scholarship 2008/54708-6 and grant 2006/58850-6), CAPES (Pró-Engenharias), and CNPq (research fellowships).