Research Article  Open Access
David Sotelo, Antonio FavelaContreras, Viacheslav V. Kalashnikov, Carlos Sotelo, "Model Predictive Control with a Relaxed Cost Function for Constrained Linear Systems", Mathematical Problems in Engineering, vol. 2020, Article ID 7485865, 10 pages, 2020. https://doi.org/10.1155/2020/7485865
Model Predictive Control with a Relaxed Cost Function for Constrained Linear Systems
Abstract
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.
1. Introduction
Model Predictive Control (MPC) for linear systems is now a wellestablished discipline providing stability, feasibility, and robustness [1–6]. Due to its inherent ability to take into account constraints and deal with multiinput multioutput variables [7–10], it has been applied in a wide range of applications, including chemical processes, industrial systems, energy, health, environment, and aerospace [11–16]. In [17], a robust MPC strategy is presented to handle the trajectory tracking problem for an underactuated twowheeled inverted pendulum vehicle. Moreover, based on an MPC scheme, in [18], a control strategy is designed to an unmanned aerial vehicle for its automatic carrier landing system. Nevertheless, the computation complexity makes the multivariable MPC ineffectual for high speed applications where the controller must execute in a few milliseconds [19–23]. Moreover, the problem becomes much more complicated solving such an online constrained optimization problem by computing a numerical solver [24–26]. Several MPC techniques are used to overcome these problems. For instance, in [27], an explicit model predictive control moves major part of computation offline, which makes it enable to be implemented in real time for wide range of fast systems. Also, in [28], to reduce the online computational time, all the state trajectories are included in the optimal control problem as the constraints in the prediction horizon, then only a quadratic programming problem is solved. In [29], based on a mixed integer quadratic programming problem, the control input is calculated at each discrete time. In contrast to common MPC approaches, where an optimization toolbox is required, this work presents a relaxed performance index, in which the weighting matrices are computed online using the concept of Taylor series expansion and standard inverse distance weighting (IDW) functions. Then, tracking performance under inputoutput constraints is well obtained, lighter computation load is achieved, and execution time to solve a Quadratic Program (QP) is reduced. Thus, a computationally efficient constrained MPC for discretetime statespace multivariable systems is obtained.
The paper is organized as follows. Section 2 gives the preliminaries of the proposed MPC strategy. Section 3 describes the proposed relaxed cost function. Section 4 presents a tape transport system and a jet transport aircraft as study cases. Simulation results show the performance of the proposed MPC strategy and the execution time improvement compared with two wellknown MPC strategies. Finally, Section 5 discusses the conclusions. Acknowledgments and the list of references finish the paper.
2. Model Predictive Control Based on DiscreteTime State Space Model
This section presents a brief review of MPC based on discretetime statespace model. The original controller is proposed by Alamir in [7]. In this previous work, considering the predictions of the states, the control action is obtained through the solution of a constrained optimization problem by using a cost function with constant weighting matrices. At each sampling time, an optimal control problem is solved whose results are computationally expensive. The system dynamics is denoted by the Linear Time Invariant (LTI) StateSpace Model taking the following structure:where is the state vector, is the controlled input vector, is the output vector, is the state matrix, stands for the input matrix, is the output matrix, and denotes the sampling instant number. Hence, as in [7], from (1), the state predictions for the consecutive sampling instants arewhere and are used to represent the Nstepahead prediction map for Linear Time Invariant (LTI) systems in a compactness form (2). Thus, system (2) can be reformulated using the following vectormatrix notation:whereand is the whole state trajectory of , concatenates the computed sequence of , and is the output trajectory of :
Now, any candidate sequence of actions has the corresponding future behavior of the system contained in the state trajectory and consequently the output trajectory .
Defining the projection matrix ,where is the vector length, corresponds to the prediction horizon, and stands for the desired term, then the state, control, and output vectors at a specific instant can be obtained as follows:
3. Proposed Relaxed Cost Function
3.1. Development
To find the best sequence of control action, in the present work, the value of the cost function is defined over the prediction horizon as follows:where considering a past control action value , the last two terms are added to penalize the rate excursion of the control vector by the weighting matrix . Moreover, in the first term, the pondering matrix penalizes the error between the output trajectory and the reference of the vector , which is expressed as follows:
Moreover, substituting (1) and (2) in the first element of (10), the cost function is expressed in statespace representation:
Thus, to obtain the control law, the performance index (11) is rewritten in the following compact form:where
Then, the control law is obtained from (12) as follows:and applying the first action of the best sequence control action , the MPC state feedback is
The MPC optimization problem (16) is solved at each sampling instant in which the measurements of the outputs and state variables are updated continuously:
3.2. Definition of the Constraints
The systematic handling of constraints provided by predictive control strategies allows for significant improvements in performance over conventional control methodologies [30]. As in [7, 31], the sequence of the future actions cannot generally be freely chosen in . At least, saturation constraints on the actuators have to be taken into account giving the following optimization problem :
The cost function is minimized considering the constraints arranged in . Nevertheless, solving general optimization problems involving a high number of decision variables and high number of constraints is generally a very difficult task. For this reason, in the present work, based on the Inverse Distance Weighting (IDW) method [24, 32–36], the cost function (12) input and output constraints are included in and matrices, where, to reduce computational complexity, Taylor series expansion is used to obtain the predicted manipulation and predicted output [23, 37–41]:
The penalization at the output is defined:where , is the tuning parameter set for a desired closedloop performance, , and , which corresponds to an IDW function.
The deviation between the predicted output and the reference is penalized by a function of distance . Furthermore, to avoid that the predicted process variable remains close to the lower bound or the upper bound the term is added. Finally, term is included in order to increase the penalization while the predicted manipulation variable is close to the upper or lower bounds, and , respectively.
The penalization at the input is defined:where .
The first term is included to avoid that the predicted manipulated variable remains close to the upper or lower bounds, and , while the second term is added to reduce the large excursion between the increment of the predicted manipulated variable and the increment of the previous manipulated variable .
The resulting optimization problem expressed as the cost function is minimized at each instant by a sequence of future actions respecting all the constraints. As it is described, and are defined as functionals of constraints depending on the current state , the desired trajectory , and the past controlled input . Thus, the matrices , , are computed online looking for relaxing the optimization problem.
3.3. Algorithm for the Proposed MPC Strategy
The algorithm used for the proposed discretetime statespace constrained MPC strategy consists of(1)Define the linear time invariant mathematical model of the physical system in statespace representation (equation (1))(2)Compute the matrices and used to represent the Nstepahead prediction map for LTI systems (equation (4))(3)Estimate the manipulation (equation (18)) and the output (equation (19)) at the next sampling instant, using Taylor series expansion(4)Compute the weighting matrices (equation (20)) and (equation (19)) based on the proposed functions(5)Compute the online matrices , , of the proposed performance index (equation (12))(6)Obtain the control law (equation (14)) and apply to the system the first action of the best sequence control action (equation (15))(7)At the next sampling instant , the reference , the bounds , , , and , the measurements of the outputs , and the state variables are updated, and the MPC optimization problem is solved again (equation (17)); the algorithm goes to step 3
4. Simulation and Results
The present work and the MPC strategies in [7, 31] are developed using the same computational platform for its evaluation. Hence, to solve the optimization problem in [7, 31], quadprog toolbox from MATLAB® is used. MATLAB® codes for the following study cases are available: MPC Matlab Files_MPE.
4.1. Example 1: Tape Transport System
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 1 shows the schematic of a tape transport system where its components and variables involved are the tape stiffness and the damping denoted by and , the reel radii and the inertia represented as and , the motor torque constant , and the viscous friction coefficient denoted by .
Assuming there is no force loss across the head, the tape tension [42]. Although the physical model of the process contains nonlinearities, in (22), a simplified statespace model is presented in continuous time [42–44]:
The model has three states, , where is the tension tape in ; meanwhile, and represent the supply and takeup reel in , respectively. Moreover, the system has two inputs that represent the voltages applied to the reel motors in , and two outputs which stand for the tape speed at the readwrite head in and the tape tension , respectively. The control strategy described in the present work is simulated using parameters from the tested tape system described in [42, 45, 46], whose parameters are summarized in Table 1. Considering that the motors are nominally identical, for both motors, it is used as the same motor torque constant and viscous friction coefficient [45, 46].

Discretizing (23) with a sampling time and considering a prediction horizon with a tuning parameter , the simulation results are shown in Figure 2. It is divided in two main parts, the first part corresponds to the outputs and the inputs of the system using the MPC with the relaxed cost function, while the second part shows the coefficient values of the weighting matrices and . As it is shown, computing the coefficient values and , the constrained tape speed and tape tension present a good performance. Here, do not present overshoot and has a maximum settling time of 0.1 seconds while has a maximum overshoot of 3% and a settling time of 0.3 seconds. On the contrary, the motor voltages for the supply reel and for the takeup reel remains inside their lower and upper bounds by using the computed coefficient values and . Here, the values of are lower than the values of , due to the rates of the manipulations.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
In order to see the closedloop stability of the system, the stability indicator is defined as in [7]:where stands for the eigenvalues of the system for and is the closedloop gain. Figure 3 shows that the system remains stable during the test. Finally, Table 2 is presented to compare the execution time between the present work and previous works [7, 31].
As it can be seen, the total execution time is reduced, by taking advantage of the relaxed cost function. Here, the computation of is 1.924 seconds and 2.148 seconds faster than the computation of the manipulations using [7, 31] MPCs strategies. Henceforth, the percentage consumption of time to obtain the control actions is notably decreased.
4.2. Example 2: Jet Transport Aircraft
The Jet Transport Aircraft Boeing 747 in highlift configuration addresses complex geometries and physical phenomena that make the controller design a difficult process. Figure 4 illustrates the Jet Transport Aircraft with its components and variables involved such as the angles and and the angular velocities and .
Although the physical model of the Boeing 747 is lengthy, in (24), the simplified statespace model during cruise flight at and is presented in continuous time [47]:
The model has four states, , where is the sideslip angle, stands for the bank angle, and meanwhile and represent the yaw and roll rate, respectively. Herein, all the angles are in and the angular velocities in . The system has two inputs : the rudder and the aileron deflections, and two outputs : the yaw rate and the bank angle .
Using a sampling time , system (24) is discretized. Then, a set of changes in the output reference and a series of variations in the constraints are used to test the present work. Considering a prediction horizon and a tuning parameter , simulation results are shown in Figure 5. Here, using the computed coefficient values and , the yaw rate has a maximum overshoot of 2.9% with a maximum settling time of 1 second, while the bank angle has a maximum overshoot of 6.6% with a maximum settling time of 1.2 seconds. Moreover, with the computed coefficient values and , the rudder deflection , and the aileron deflection remains inside their bounds without saturation. Here, the values of are greater than the values of , this is because the rate of the manipulation is greater than the rate of the manipulation . On the contrary, Figure 6 shows the stability behavior during the test. As it is shown, the system remains stable.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Finally, the execution time comparison between the present work and previous works [7, 31] is shown in Table 3. As it can be seen, the total execution time is considerably reduced due to the relaxed cost function. Here, the computation of is 6.074 seconds and 3.155 seconds faster than the computation of the manipulations using [7, 31] MPCs strategies. Then, as in example 1, the percentage consumption of time to obtain the control actions is notably decreased.
5. Conclusions
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 is 1.3% for the tape transport system and 1.7% for the jet transport aircraft. Henceforth, it takes almost 0.1 milliseconds for the tape transport system and 0.12 milliseconds for the jet transport aircraft to obtain the manipulation that minimizes the proposed cost function while respecting the constraints. Thus, the proposed MPC strategy with the relaxed cost function is mainly suitable for constrained multivariable real processes with fast dynamics.
Data Availability
The data of the conducted experiments and simulations are available upon requirement.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
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.
References
 R. Heydari and M. Farrokhi, “Robust model predictive control of biped robots with adaptive online gait generation,” International Journal of Control, Automation and Systems, vol. 15, no. 1, pp. 329–344, 2017. View at: Publisher Site  Google Scholar
 S. Riverso, M. Farina, and G. FerrariTrecate, “Plugandplay decentralized model predictive control for linear systems,” IEEE Transactions on Automatic Control, vol. 58, no. 10, pp. 2608–2614, 2013. View at: Publisher Site  Google Scholar
 R. Zhang, J. Lu, H. Qu, and F. Gao, “State space model predictive faulttolerant control for batch processes with partial actuator failure,” Journal of Process Control, vol. 24, no. 5, pp. 613–620, 2014. View at: Publisher Site  Google Scholar
 M. Zhao, C.C Jiang, and M.H. She, “Robust contractive economic MPC for nonlinear systems with additive disturbance,” International Journal of Control, Automation and Systems, vol. 16, no. 5, pp. 2253–2263, 2018. View at: Publisher Site  Google Scholar
 H. Shi, P. Li, J. Cao, C. Su, and J. Yu, “Robust fuzzy predictive control for discretetime systems with interval timevarying delays and unknown disturbances,” IEEE Transactions on Fuzzy Systems, 2019. View at: Publisher Site  Google Scholar
 H. Shi, P. Li, C. Su, Y. Wang, J. Yu, and J. Cao, “Robust constrained model predictive faulttolerant control for industrial processes with partial actuator failures and interval timevarying delays,” Journal of Process Control, vol. 75, pp. 187–203, 2019. View at: Publisher Site  Google Scholar
 M. Alamir, A Pragmatic Story of Model Predictive Control: SelfContained Algorithms and CaseStudies, CNRSUniversity of Grenoble, Grenoble, France, 2013.
 I. Chang and J. Bentsman, “Constrained discretetime statedependent Riccati equation technique: a model predictive control approach,” in Proceedings of the of 52nd IEEE Conference on Decision and Control, pp. 5125–5130, Florence, Italy, December 2013. View at: Publisher Site  Google Scholar
 B. Zhu, H. Tazvinga, and X. Xia, “Switched model predictive control for energy dispatching of a photovoltaicdieselbattery hybrid power system,” IEEE Transactions on Control Systems Technology, vol. 23, no. 3, pp. 1229–1236, 2015. View at: Publisher Site  Google Scholar
 H.y. Shi, C.l. Su, J.t. Cao, P. Li, Y.l. Song, and N.b. Li, “Incremental multivariable predictive functional control and its application in a gas fractionation unit,” Journal of Central South University, vol. 22, no. 12, pp. 4653–4668, 2015. View at: Publisher Site  Google Scholar
 Y. Wang and S. Boyd, “Fast model predictive control using online optimization,” IEEE Transactions on Control Systems Technology, vol. 18, no. 2, pp. 267–278, 2010. View at: Publisher Site  Google Scholar
 S.K. Kim, D.K. Choi, K.B. Lee, and Y. I. Lee, “Offsetfree model predictive control for the power control of threephase AC/DC converters,” IEEE Transactions on Industrial Electronics, vol. 62, no. 11, pp. 7114–7126, 2015. View at: Publisher Site  Google Scholar
 R. Zhang, A. Xue, and F. Gao, “Temperature control of industrial coke furnace using novel state space model predictive control,” IEEE Transactions on Industrial Informatics, vol. 10, no. 4, pp. 2084–2092, 2014. View at: Publisher Site  Google Scholar
 M. Preindl and S. Bolognani, “Model predictive direct speed control with finite control set of PMSM drive systems,” IEEE Transactions on Power Electronics, vol. 28, no. 2, pp. 1007–1015, 2013. View at: Publisher Site  Google Scholar
 B. Hredzak, V. G. Agelidis, and M. Minsoo Jang, “A model predictive control system for a hybrid batteryultracapacitor power source,” IEEE Transactions on Power Electronics, vol. 29, no. 3, pp. 1469–1479, 2014. View at: Publisher Site  Google Scholar
 H. Shi, C. Su, J. Cao, P. Li, J. Liang, and G. Zhong, “Nonlinear adaptive predictive functional control based on the TakagiSugeno model for average cracking outlet temperature of the ethylene cracking furnace,” Industrial & Engineering Chemistry Research, vol. 54, no. 6, pp. 1849–1860, 2015. View at: Publisher Site  Google Scholar
 M. Yue, C. An, and J.Z. Sun, “An efficient model predictive control for trajectory tracking of wheeled inverted pendulum vehicles with various physical constraints,” International Journal of Control, Automation and Systems, vol. 16, no. 1, pp. 265–274, 2018. View at: Publisher Site  Google Scholar
 S. Koo, S. Kim, J. Suk, Y. Kim, and J. Shin, “Improvement of shipboard landing performance of fixedwing UAV using model predictive control,” International Journal of Control, Automation and Systems, vol. 18, no. 1, pp. 265–274, 2018. View at: Google Scholar
 R. Zhang, S. Wu, and F. Gao, “State space model predictive control for advanced process operation: a review of recent development, new results, and insight,” Industrial & Engineering Chemistry Research, vol. 56, no. 18, pp. 5360–5394, 2017. View at: Publisher Site  Google Scholar
 X. YuGeng, L. DeWei, and L. Shu, “Model predictive control—status and challenges,” Acta Automatica Sinica, vol. 39, no. 3, pp. 222–236, 2013. View at: Publisher Site  Google Scholar
 S. J. Qin and T. A. Badgwell, “A survey of industrial model predictive control technology,” Control Engineering Practice, vol. 11, no. 7, pp. 733–764, 2003. View at: Publisher Site  Google Scholar
 P. D. Christofides, R. Scattolini, D. Muñoz de la Peña, and J. Liu, “Distributed model predictive control: a tutorial review and future research directions,” Computers & Chemical Engineering, vol. 51, pp. 21–41, 2013. View at: Publisher Site  Google Scholar
 J. Yang, W. X. Zheng, S. Li, B. Wu, and M. Cheng, “Design of a predictionaccuracyenhanced continuoustime MPC for disturbed systems via a disturbance observer,” IEEE Transactions on Industrial Electronics, vol. 62, no. 9, pp. 5807–5816, 2015. View at: Publisher Site  Google Scholar
 A. Bemporad, “Global optimization via inverse distance weighting,” 2019, https://arxiv.org/abs/1906.06498. View at: Google Scholar
 A. Bemporad, “Model predictive control design: new trends and tools,” in Proceedings of the 45th IEEE Conference on Decision and Control, pp. 6678–6683, San Diego, CA, USA, December 2006. View at: Publisher Site  Google Scholar
 A. Bemporad, “A multiparametric quadratic programming algorithm with polyhedral computations based on nonnegative least squares,” IEEE Transactions on Automatic Control, vol. 60, no. 11, pp. 2892–2903, 2015. View at: Publisher Site  Google Scholar
 M. A. Mohammadkhani, F. Bayat, and A. A. Jalali, “Design of explicit model predictive control for constrained linear systems with disturbances,” International Journal of Control, Automation and Systems, vol. 12, no. 2, pp. 294–301, 2014. View at: Publisher Site  Google Scholar
 M. A. Mousavi, B. Moshiri, and Z. Heshmati, “A new predictive motion control of a planar vehicle under uncertainty via convex optimization,” International Journal of Control, Automation and Systems, vol. 15, no. 1, pp. 129–137, 2017. View at: Publisher Site  Google Scholar
 D. Satoh, K. Kobayashi, and Y. Yamashita, “MPCbased codesign of control and routing for wireless sensor and actuator networks,” International Journal of Control, Automation and Systems, vol. 16, no. 3, pp. 953–960, 2018. View at: Publisher Site  Google Scholar
 G. Serale, M. Fiorentini, A. Capozzoli, D. Bernardini, and A. Bemporad, “Model predictive control (MPC) for enhancing building and HVAC system energy efficiency: problem formulation, applications and opportunities,” Energies, vol. 11, no. 3, p. 631, 2018. View at: Publisher Site  Google Scholar
 L. Wang, Model Predictive Control System Design and Implementation Using MATLAB®, Springer Science & Business Media, Berlin, Germany, 2009.
 Z. Li, K. Wang, H. Ma, and Y. Wu, “An adjusted inverse distance weighted spatial interpolation method,” in Proceedings of the 2018 3rd International Conference on Communications, Information Management and Network Security (CIMNS 2018), Atlantis Press, Wuhan, China, November 2018. View at: Publisher Site  Google Scholar
 Z. Fan, J. Li, and M. Deng, “An adaptive inversedistance weighting spatial interpolation method with the consideration of multiple factors,” Geomatics and Information Science of Wuhan University, vol. 6, 20 pages, 2016. View at: Google Scholar
 D. Shepard, “A twodimensional interpolation function for irregularlyspaced data,” in Proceedings of the of the 23rd ACM National Conference, pp. 517–524, New York, NY, USA, 1968. View at: Publisher Site  Google Scholar
 G. Y. Lu and D. W. Wong, “An adaptive inversedistance weighting spatial interpolation technique,” Computers & Geosciences, vol. 34, no. 9, pp. 1044–1055, 2008. View at: Publisher Site  Google Scholar
 D. W. Wang, L. N. Li, C. Hu, Q. Li, X. Chen, and P. W. Huang, “A modified inverse distance weighting method for interpolation in open public places based on wifi probe data,” Journal of Advanced Transportation, Article ID 7602792, 11 pages, 2019. View at: Publisher Site  Google Scholar
 M. Ławryńczuk, “Nonlinear predictive control of a boilerturbine unit: a statespace approach with successive online model linearisation and quadratic optimisation,” ISA Transactions, vol. 67, pp. 476–495, 2017. View at: Publisher Site  Google Scholar
 M. Ławryńczuk, “Nonlinear state–space predictive control with on–line linearisation and state estimation,” International Journal of Applied Mathematics and Computer Science, vol. 25, no. 4, pp. 833–847, 2015. View at: Publisher Site  Google Scholar
 W. H. Chen, “Predictive control of general nonlinear systems using approximation,” IEE Proceedings—Control Theory and Applications, vol. 151, no. 2, pp. 137–144, 2004. View at: Publisher Site  Google Scholar
 R. Errouissi, A. AlDurra, and S. M. Muyeen, “A robust continuoustime MPC of a DCDC boost converter interfaced with a gridconnected photovoltaic system,” IEEE Journal of Photovoltaics, vol. 6, no. 6, pp. 1619–1629, 2016. View at: Publisher Site  Google Scholar
 C. Sotelo, A. FavelaContreras, F. BeltránCarbajal, G. DieckAssad, P. RodríguezCañedo, and D. Sotelo, “A novel discretetime nonlinear model predictive control based on state space model,” International Journal of Control, Automation and Systems, vol. 16, no. 6, pp. 2688–2696, 2018. View at: Publisher Site  Google Scholar
 P. D. Mathur and W. C. Messner, “Controller development for a prototype highspeed lowtension tape transport,” IEEE Transactions on Control Systems Technology, vol. 6, no. 4, pp. 534–542, 1998. View at: Publisher Site  Google Scholar
 Y. LU and W. C. Messner, “Robust servo design for tape transport,” in Proceedings of the 2001 IEEE International Conference on Control Applications (CCA’01) (Cat. No. 01CH37204), pp. 1014–1019, Mexico City, Mexico, September 2001. View at: Publisher Site  Google Scholar
 D. Tenne and T. Singh, “Robust feedforward/feedback design for tape transport,” in Proceedings of the of the AIAA Guidance, Navigation, and Control Conference and Exhibit, p. 5119, Providence, RI, USA, August 2004. View at: Publisher Site  Google Scholar
 M. D. Baumgart and L. Y. Pao, “Robust control of nonlinear tape transport systems with and without tension sensors,” Journal of Dynamic Systems, Measurement, and Control, vol. 129, no. 1, pp. 41–55, 2007. View at: Publisher Site  Google Scholar
 M. D. Baumgart and L. Y. Pao, “Robust control of tape transport systems with no tension sensor,” in Proceedings of the 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No. 04CH37601), pp. 4342–4349, Nassau, Bahamas, December 2004. View at: Publisher Site  Google Scholar
 S. Singh and T. R. Murthy, “Simulation of sensor failure accommodation in flight control system of transport aircraft: a modular Approach,” World Journal of Modelling and Simulation, vol. 11, no. 1, pp. 55–68, 2015. View at: Google Scholar
Copyright
Copyright © 2020 David Sotelo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.