**Model Predictive Control**

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Password Changed Successfully Your password has been changed. Returning user. In the simulations presented here the following sequence is adopted:. The output of this model is designated [y p ] k. This observer is based on the assumption that the difference between model output and plant measurement is produced by a step disturbance in the system output.

### Original Research ARTICLE

This assumption is adopted by most MPC packages but is restricted to stable systems, which is the case considered here. Therefore, assuming that the uncertain process belongs to W , one estimates the states of all the linear models lying in W using 13 and U defines the set of feasible solutions related to the input constraints.

Problem P1 can be solved by the available methods of linear matrix inequalities LMI. Consider an open-loop stable process where the true plant model belongs to W and whose states are estimated by a stable observer. If there is a feasible solution to Problem P1 , then the resulting control law stabilizes all the plants belonging to W. The proof for the state feedback case can be found in Rodrigues and Odloak It remains to be proved that when a stable state observer is used the resulting closed loop remains stable.

Thus, as a consequence of the separation principle, when one uses IHMPC, which stabilizes all the models belonging to set W when the states are perfectly known, and a stable state observer, the resulting closed-loop will also be stable. The approach is based on the parameterization of the control moves by an output feedback scheme similar to the conventional linear quadratic regulator.

In this way, it was possible to use the quadratic Lyapunov stability condition that was included in MPC as an additional constraint.

In their MPC, these authors considered a polytopic representation of the uncertain models. The multimodel version of the Rodrigues and Odloak robust predictive controller is described below:. Consequently, the dimension of the state x of this controller is different from the dimension of the state considered by RIHMPC.

The process simulated in this work is a debutanizer column of an oil refinery, and it was borrowed from Rodrigues and Odloak This column produces LPG liquefied petroleum gas as the top stream and stabilized gasoline as the bottom stream. The manipulated inputs are the top reflux flow rate u 1 and the reboiler heat duty u 2. The set of uncertain models, W , was built by linear models identified at three different operating points of the column. These models can be seen in Table 1. The most probable model is assumed to be G 1 s.

The case of output tracking where the reference values of the two outputs were simultaneously increased is considered.

In this paper an extension of a highly numerical efficient robust infinite horizon MPC was developed for the output feedback case. A case study using an industrial process was carried out in order to compare the proposed controller to another robust MPC presented earlier in the MPC literature. Although RIHMPC is a controller where a min-max problem is solved, and consequently the worst case performance is considered, its closed loop performance is better then the performance of RSMPC whose control law is based on the most probable system model.

Consequently, inclusion of the Lyapunov stability condition in the MPC problem can make the controller more conservative than the controller based on the min-max approach.

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## Robust shrinking ellipsoid model predictive control for linear parameter varying system

In this case, stability is guaranteed for any plant whose model is a convex combination of the models of W. Badgwell, T. International Journal of Control, 68, Automatica, 25, 3, Automatica, 32, 10, Robust predictive control of non-linear systems under state estimation errors and input and state constraints is a challenging problem, and solutions to it have generally involved solving computationally hard non-linear optimizations.

Feedback linearization has reduced the computational burden, but has not yet been solved for robust model predictive control under estimation errors and constraints.

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In this paper, we solve this problem of robust control of a non-linear system under bounded state estimation errors and input and state constraints using feedback linearization. We do so by developing robust constraints on the feedback linearized system such that the non-linear system respects its constraints.

These constraints are computed at run-time using online reachability, and are linear in the optimization variables, resulting in a Quadratic Program with linear constraints. We also provide robust feasibility, recursive feasibility and stability results for our control algorithm.