The Model Predictive Control technique is widely used for optimizing the performance of constrained multiinput multioutput processes. However, due to its mathematical complexity and heavy computation effort, it is mainly suitable in processes with slow dynamics. Based on the Exact Penalization Theorem, this paper presents a discretetime statespace Model Predictive Control strategy with a relaxed performance index, where the constraints are implicitly defined in the weighting matrices, computed at each sampling time. The performance validation for the Model Predictive Control strategy with the proposed relaxed cost function uses the simulation of a tape transport system and a jet transport aircraft during cruise flight. Without affecting the tracking performance, numerical results show that the execution time is notably decreased compared with two wellknown discretetime statespace Model Predictive Control strategies. This makes the proposed Model Predictive Control mainly suitable for constrained multivariable processes with fast dynamics.
Model Predictive Control (MPC) for linear systems is now a wellestablished discipline providing stability, feasibility, and robustness [
The paper is organized as follows. Section
This section presents a brief review of MPC based on discretetime statespace model. The original controller is proposed by Alamir in [
Now, any candidate sequence of actions
Defining the projection matrix
To find the best sequence of control action, in the present work, the value of the cost function
Using notation (
Moreover, substituting (
Thus, to obtain the control law, the performance index (
Then, the control law is obtained from (
The MPC optimization problem (
The systematic handling of constraints provided by predictive control strategies allows for significant improvements in performance over conventional control methodologies [
The cost function
The penalization at the output
The deviation between the predicted output
The penalization at the input
The first term is included to avoid that the predicted manipulated variable
The resulting optimization problem
The algorithm used for the proposed discretetime statespace constrained MPC strategy consists of
Define the linear time invariant mathematical model of the physical system in statespace representation (equation (
Compute the matrices
Estimate the manipulation
Compute the weighting matrices
Compute the online matrices
Obtain the control law
At the next sampling instant
The present work and the MPC strategies in [
The tape drive system consists of two reels to supply and file data. Here, the data transfer rate is proportional to the tape transport speed. Thus, the tape drive mechanism must be able to rapidly transport a fragile tape with an accurate tension regulation. Figure
Tape transport system.
Assuming there is no force loss across the head, the tape tension
The model has three states,
Tape transport system parameters.
Symbol  Parameter  Value 


Tape stiffness  2 × 10^{3} N/m 

Damping  2 N s/m^{2} 

Radius of supply reel  21.2 × 10^{−3} m 

Radius of takeup reel  9.75 × 10^{−3} m 

Moment of inertia of the supply reel  14.2 × 10^{−6} Kg m^{2} 

Moment of inertia of the takeup reel  10.35 × 10^{−6} Kg m^{2} 

Motor torque constant  24.8 × 10^{−3} N m/V 

Viscous friction coefficient  1.03 × 10^{−4} N m s/rad 
Discretizing (
Simulation results for tape transport system under the MPC strategy with the relaxed cost function.
In order to see the closedloop stability of the system, the stability indicator
Stability behavior.
Execution time comparison with previous works.
MPC  Total (s)  Computation of 
Percentage consumption (%) 

[ 
2.919  1.936  66.3 
[ 
2.907  2.160  74.3 
This work  0.892  0.012  1.3 
As it can be seen, the total execution time is reduced, by taking advantage of the relaxed cost function. Here, the computation of
The Jet Transport Aircraft Boeing 747 in highlift configuration addresses complex geometries and physical phenomena that make the controller design a difficult process. Figure
Jet transport aircraft. (a) Top view. (b) Front view.
Although the physical model of the Boeing 747 is lengthy, in (
The model has four states,
Using a sampling time
Simulation results for jet transport aircraft under the MPC strategy with the relaxed cost function.
Stability behavior.
Finally, the execution time comparison between the present work and previous works [
Execution time comparison with previous works.
MPC  Total (s)  Computation of 
Percentage consumption (%) 

[ 
8.262  6.105  73.9 
[ 
4.020  3.186  79.2 
This work  1.808  0.031  1.7 
This paper presents a discretetime statespace MPC approach for multivariable systems. Based on the IDW method and the concept of Taylor series expansion, a relaxed performance index with constraints defined in the online weighting matrices is proposed to compute the control action. Thus, as in study cases, the proposed MPC strategy is used to control a tape transport system and a jet transport aircraft during cruise flight.
Simulation results show that the proposed MPC strategy with the relaxed cost function has a good performance, no matter abrupt changes of setpoints and constraints occur, even at the same time. Additionally, compared with two wellknown discretetime statespace MPC strategies, there is a significant improvement on the execution time without affecting the tracking performance. The percentage consumption of time to compute the best sequence of control actions
The data of the conducted experiments and simulations are available upon requirement.
The authors declare that they have no conflicts of interest.
The research activities of the third coauthor were partially supported by the Mexico National Council for Science and Technologies (CONACYT) with Grants CB201301221676 and FC2016011938. The authors also thank the Research Groups of Sensors and Devices, and of Optimization and Data Science of the School of Engineering and Sciences for their support of the development of this work and MSc. Arturo Pinto for his fruitful discussions.