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Shock and Vibration
Volume 2014 (2014), Article ID 324018, 12 pages
http://dx.doi.org/10.1155/2014/324018
Research Article

Robust Tracking Control for Vehicle Lateral Dynamics with Uncertain Parameters and External Nonlinearities

The State Key Laboratory of Robotics and System, Harbin Institute of Technology (HIT), Harbin 150001, China

Received 16 September 2012; Accepted 12 March 2013; Published 5 March 2014

Academic Editor: Hongyi Li

Copyright © 2014 Huihui Pan 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.

Abstract

This paper focuses on the problem of tracking control for vehicle lateral dynamic systems and designs an adaptive robust controller (ARC) based on backstepping technology to improve vehicle handling and stability, in the presence of parameter uncertainties and external nonlinearities. The main target of controller design has two aspects: the first target is to control the sideslip angle as small as possible, and the second one is to keep the real yaw rate tracking the desired yaw rate. In order to compromise the two indexes, the desired sideslip angle is planned as a new reference signal, instead of the ideal “zero.” As a result, the designed controller not only accomplishes the control purposes mentioned above, but also effectively attenuates both the changes of vehicle mass and the variations of cornering stiffness. In addition, to overcome the problem of “explosion of complexity” caused by backstepping method in the traditional ARC design, the dynamic surface control (DSC) technique is used to estimate the derivative of the virtual control. Finally, a nonlinear vehicle model is employed as the design example to illustrate the effectiveness of the proposed control laws.

1. Introduction

Because of the large number of traffic accidents occurring daily, vehicle safety control has been a hot research topic. In recent years, vehicle lateral dynamic control has been studied widely for contributing to the car’s handling and keeping vehicle ride safe. Especially the yaw-moment control is proved to be an important approach to improve safety performance and has a great potential to meet the requirements demanded by users. To this end, a considerable amount of research has been carried out [112], and many vehicle lateral control approaches have been proposed, based on various control techniques, such as fuzzy logic control [1315], control [16], adaptive control [1720], and nonlinear robust control [21, 22]. Such strategies could considerably enhance vehicle handling and active safety during severe driving maneuvers and, at the same time, allow the driver to keep control of the vehicle when the vehicle is at the physical limit of adhesion between the tires and the road.

In vehicle dynamic systems, inevitable uncertainties often emerge, which will bring considerable difficulties in the process of controller design. For example, because of the change of the number of passengers or the payload, vehicle load is easily varied, which will accordingly change the vehicle mass as a varying parameter. On the other side, the moment of inertia is usually an unknown parameter. Besides, since the yaw-moment control relies on the tire lateral force and the tire force strongly depends on tire vertical load, which is very sensitive to vehicle motion and environmental conditions, the tire cornering stiffness inevitably obtains uncertainties that need to be coped with. Roughly speaking, the abovementioned uncertainties can be classified into two categories: parametric uncertainties (e.g., car body mass for vehicle dynamic control) and general uncertainties (coming from modeling errors and external disturbances). To handle this situation, a number of control techniques have been proposed, such as robust or control [2326], optimal control [27], fuzzy control [28, 29], sliding mode control [30, 31], neural network control [3238], adaptive control [39], and fault tolerant control [40, 41]. Besides, during the past decade, a mathematically rigorous nonlinear adaptive robust control theory has also been developed to lay a solid foundation for the design of a new generation of controllers which will help industry build modern machines of great performance and high intelligence [4244]. This ARC approach can be both adaptive to the uncertain parameters and at the same time robust against the external disturbances, which is suitable for the systems with uncertainties and the external disturbances, for example, vehicle dynamic systems and robot manipulation systems [45].

The main target of yaw-moment control can be divided into two aspects. The first target is to control the sideslip angle to converge to zero, and the second one is to keep the real yaw rate tracking the desired trajectory. However, these two requirements are conflicting, and it is difficult to achieve both these two indexes simultaneously, especially for the systems with uncertainties. Therefore, a compromise of the requirements must be reached. In this paper, the problem of tracking control for vehicle lateral dynamic systems is investigated, and a backstepping-type adaptive robust controller is designed to improve vehicle handling and stability, in the presence of parameter uncertainties and external nonlinearities. In order to compromise the two tracking indexes, the desired sideslip angle is planned as a new reference signal, instead of the ideal “zero.” As a result, the designed controller not only accomplishes the required control purposes, but also effectively attenuates both the changes of vehicle mass and the variations of cornering stiffness. In addition, to overcome the problem of “explosion of complexity” caused by backstepping method in the traditional ARC design, the DSC technique is used to estimate the derivative of the virtual control. Furthermore, the adaptive law is designed to estimate the real value of the moment of inertia. Finally, a nonlinear vehicle model is employed as the design example to illustrate the effectiveness of the proposed control law.

The rest of this paper is organized as follows. The problem to be solved is formulated mathematically in Section 2, and controller design is presented in Section 3, where both the traditional and improved ARC designs are presented. Section 4 provides a design example to illustrate the effectiveness of the proposed control laws and some concluding remarks are given in Section 5.

Nomenclature. The following nomenclature is used throughout the paper: is used to denote the estimate of is used to denote the parameter estimation error of , and , are the maximum and minimum values of for all , respectively.

2. Problem Formulation

In this paper, a “bicycle model” is used to model the dynamics of the car, as shown in Figure 1. In this figure, represents the car chassis; is the moment of inertia about the yaw axis through the center of gravity (CG). The front and rear axles are located at distances and , respectively, from the vehicle CG. The front and rear lateral tire forces and depend on slip angles and , respectively, and the steering angle changes the heading of the front tires. and stand for the sideslip angle and yaw rate, respectively, and is the external yaw moment. In this paper, it is assumed that the vehicle velocity is a constant and the steering angle is small.

324018.fig.001
Figure 1: Simplified vehicle ‘‘bicycle model.”

The equations of the vehicle’s handling dynamics in the yaw plane are given as The front and rear lateral tire forces in (1) can be given as where are the lateral forces at the front and rear wheels in case that the tires operate in the linear region and , are the additional nonlinear terms and are bounded. and are the cornering stiffness for the front and rear tires, respectively, and the front and rear slip angles are defined as Define the nonlinear disturbance uncertainty , where is used to describe the varying of the vehicle mass . Similarly, we define , and it is assumed that and , , where is a positive constant and is the uncertain parameter which satisfies . Based on the above definitions, we can rewrite the dynamic equations in (1) as where and with , being the uncertain parameter.

Problem 1. For the lateral dynamics systems in (1), synthesize a control input to control the sideslip angle as small as possible, and keep the real yaw rate tracking the desired trajectory, in the presence of parametric uncertainties and external disturbances.

3. Control Law Synthesis

3.1. Adaptive Robust Controller Design

In this section, an adaptive robust controller is presented to track the desired trajectories of the lateral dynamic systems with uncertain parameters and external disturbances, and the detailed process of controller design is given as follows.

Step 1. Choose as the virtual control and design a desired function , such that if , then the tracking error is guaranteed to converge to zero or be bounded, where is the reference trajectory.

Starting with the equation of tracking error, we have Let be an error variable representing the difference between the actual and virtual control of (7); that is, . Thus we can rewrite (7) as Then, the desired virtual control can be proposed as where is used to achieve a model compensation, is the stabilizing feedback term, is a positive design parameter, and is a robust control law designed to satisfy the following conditions: where is a design parameter which can be arbitrarily small. Basically, condition 1 of (12) shows that is synthesized to dominate the model uncertainties coming from uncertain nonlinearity , and condition 2 is to make sure that is dissipating in nature so that it does not interfere with nominal process of control part and . Then, the robust control part can be chosen as where . Then, we will show how in (13) guarantees conditions 1 and 2 in (12).

Substituting into condition 1 in (12), we have

Substituting (9)–(11) and (13) into (8) results in

Step 2. Synthesize an adaptive robust control law for , so that the error is bounded in the presence of uncertain parameter and external disturbances , .

Differentiating the error dynamics for results in where . Design the control input where is a positive design parameter and , respect the certain and uncertain parts of is a robust control law designed to satisfy the following conditions: where is a design parameter which can be arbitrarily small. To satisfy the conditions above, one can choose the nonlinear control gain as where is a function which satisfies is the estimation of , which is chosen as the projection type with the following form: and is a tunable gain and . The standard projection mapping is introduced as This projection mapping guarantees that the parameter estimate is always within the known bounds, that is, , and , for all , which enables one to show that the use of projection modification to the traditional integral type adaptation law does not interfere with the perfect learning capability of the original integral type adaptation law.

Then, substituting (17)–(23) into (16) results in The structure diagram of the adaptive robust controller design is given in Figure 2, and based on the above processing of controller design, we have the following theorem.

324018.fig.002
Figure 2: Structure diagram of the adaptive robust controller design.

Theorem 2. If is designed as (17)–(19), and the adaptive law is chosen as (23), then the following results hold.(a)In general (i.e., the system is subjected to parametric uncertainties, unmodelled uncertainties, and external disturbances), both the tracking errors and are bounded and, specifically, defining , the steady-state output tracking errors and are bounded by (b)If, after a finite time, the system is subjected to parametric uncertainties only (i.e., all the disturbances vanish after a finite time), both the tracking errors and will asymptotically converge to zero.

Proof. Firstly, the proof of statement (a) is given. Choose a positive definite function as whose derivative is given as After defining , we have which can further result in Inequality (29) implies that , which guarantees Therefore, the proof of statement (a) is completed.
If, after a finite time, the system is only subjected to parametric uncertainties, the dynamic systems can be written as Choose a positive definite function as and then we have Noticing the property of the projection mapping we have Integrating both sides of inequality from to results in which implies , and thus . Therefore, and thus By using Lyapunov-like lemma, we have as , which means that the tracking errors and converge to zero asymptotically. The proof is completed.

Remark 3. As stated above, our main target has two aspects: the first target is to control the sideslip angle to converge to zero, and the second one is to keep the real yaw rate tracking the desired trajectory. However, these two requirements are conflicting, and it is difficult to achieve both these two indexes simultaneously, especially for the systems with uncertainties and nonlinearities. Therefore, a compromise of the requirements must be reached. To handle this situation, in this paper, the desired sideslip angle is replanned as follows: where and is a positive constant. Using this trajectory in (39) to replace the desired trajectory zero, we can guarantee that the virtual input converges to the desired yaw rate , where in which is a stability factor.

Next, we will give the proof that will converge to the desired yaw rate , using the designed . Defining and choosing a positive function as we have Substituting (39) into (44) results in which means that will converge to zero as , and it further implies that coverages to the desired yaw rate .

Remark 4. Actuator saturation appears frequently in engineering systems, which is also a source of performance degradation and the closed-loop system instability. Roughly speaking, all actuators of physical devices are subject to amplitude saturation. Although, in some applications, it may be possible to ignore this fact, the reliable operation and acceptable performance of most control systems must be assessed in light of actuator saturation. From the analysis above, it is known that all the signals are bounded within the known ranges, and thus the actuator saturation constraints will be guaranteed as long as the initial values and tuning gains () are adjusted properly.

3.2. Adaptive Robust Controller Design via DSC Technique

From the design process above, it can been seen that it is quite complicated to obtain , especially to split into the known part and the unknown part . To overcome the problem of ‘‘explosion of complexity” caused by backstepping method in the traditional ARC design, the dynamic surface control technique is used to estimate the derivative of the virtual control . Therefore, the design process for Step 2 is modified as follows.

Modified Step 2. Let pass through a first-order filter with time constant , which means Defining the estimate errors as , , the derivative of can be rewritten as

Differentiating the error dynamics for results in After choosing the adaptive robust controller as we can obtain the error dynamic systems as where is a robust control law designed to satisfy the following conditions: where is a design parameter which can be arbitrarily small. Similarly, we can choose the nonlinear control gain as , where is a function which satisfies Furthermore, since all terms in can be dominated by some continuous functions , we have which implies

Before the main result is given, the following definitions are given. For any , define where Obviously, is a compact subset; hence there must be a point corresponding to the supreme value of in . We denote this supreme value as ; that is,

Theorem 5. If the virtual input, control input, and adaptive law are designed as (9), (49), and (23), respectively, then, for any initial states in , there exist positive parameters , and satisfying such that the tracking errors , and are uniformly ultimately bounded and the steady-state tracking error can be made arbitrarily small.

Proof. Define a positive function as shown in (56). The derivative of is Noting that we have Based on conditions (58), we have where with . Inequality (62) can further result in which implies that , and are bounded as
Next, we will show that the steady-state tracking errors , and can be made arbitrarily small. Define a positive definite function satisfying In order to make a contradiction, we assume that there exists so that, when , where and is a positive constant which can be set arbitrarily small. On the other hand, the following inequality is true: Integrating both sides of the above inequality from zero to any , we obtain Because is bounded, we have is bounded as well. It is obvious that is bounded, and then, based on Barbalat’s lemma, we have which implies Therefore, the tracking errors , and are uniformly ultimately bounded and the steady-state tracking error can be made arbitrarily small by properly choosing tuning parameters.

4. A Design Example

In this section, we provide an example to illustrate the effectiveness of the proposed approach. The vehicle model parameters are listed in Table 1.

tab1
Table 1: The model parameters of vehicle lateral systems.

The initial state values are assumed as zeros, and . The controller parameters are given in Table 2.

tab2
Table 2: The controller parameters of adaptive robust controller.

For checking the vehicle lateral dynamic performance in terms of the change of vehicle mass, three different masses are tested to illustrate the effectiveness of robust control:(S1)closed-loop systems with the vehicle mass  kg;(S2)closed-loop systems with the vehicle mass  kg;(S3)closed-loop systems with the vehicle mass  kg;(S4)open-loop systems without controller.

4.1. Periodic Demand Signal

In order to illustrate the effectiveness of the proposed control law, in this paper, it is assumed that the steering angle , and the lumped nonlinearities

The sideslip angle responses of the open-loop system, closed-loop systems with designed ARC controller for different vehicle masses are compared in Figure 3, and Figure 4 is the local enlargement of Figure 3 from −0.5 to 0.5 at y-axis. From these two figures, we can see that the response peak has been reduced substantially, by using the proposed controller, compared with the open-loop system without controller. Also, we can get the conclusion that the sideslip angle performance is still guaranteed at a high level, although there are changing vehicle masses, external disturbances, and other uncertainties.

324018.fig.003
Figure 3: The sideslip angle responses of the open-loop system, closed-loop systems with designed ARC controller for different vehicle masses.
324018.fig.004
Figure 4: The local enlargement of Figure 3.

Figures 5 and 6 show the time histories of yaw rate for both open-loop systems and closed-loop systems with ARC controller in the case of different vehicle masses, and Figures 7 and 8 are the corresponding responses of tracking errors between the real yaw rate and the desired yaw rate. It can be seen that the closed-loop systems with the proposed ARC controllers can track the desired trajectories perfectly. By contrast, the open-loop systems without controller cannot do this work, and the tracking errors of the open-loop systems are very large. From Figures 5 and 6, we observe that the yaw rate response of the controlled system is better than the uncontrolled system, regardless of the change of vehicle mass. In particular, from Figures 7 and 8, the tracking errors are close to zero, and the yaw rates are all smaller than the corresponding uncontrolled system responses.

324018.fig.005
Figure 5: The yaw rate responses of the open-loop system, closed-loop systems with designed ARC controller for different vehicle masses.
324018.fig.006
Figure 6: The local enlargement of Figure 5.
324018.fig.007
Figure 7: The tracking errors between real yaw rate and desired yaw rate of the open-loop system, closed-loop systems with designed ARC controller for different vehicle masses.
324018.fig.008
Figure 8: The local enlargement of Figure 7.

The simulation result in Figure 9 characterizes the adaptation of the uncertain parameter , and from this figure we can see that the moment of inertia about the yaw axis does converge to the true value (1600 kgm2), which achieves the estimation of the uncertain parameters.

324018.fig.009
Figure 9: The moment of inertia about the yaw axis, .

The control inputs (the yaw moment, ) for different cases are plotted in Figure 10.

324018.fig.0010
Figure 10: Control inputs, .
4.2. Abrupt Demand Signal

Abrupt demand signal can be generally assumed as discrete events of relatively short duration and high intensity, and the corresponding function is given by Bearing an analogy with the periodic input, the sideslip angle responses of the open-loop system, closed-loop systems with designed ARC controller for different vehicle masses are plotted in Figure 11. From Figure 11, we can see that the maximal response peak is less than 0.1 m by using the proposed controller, which has been reduced substantially compared with the open-loop system without controller (the maximal response peak of open-loop systems is nearly 0.4 m). The proposed controller is also robust against the changing vehicle masses, which can be seen from Figures 1114.

324018.fig.0011
Figure 11: The sideslip angle responses of the open-loop system, closed-loop systems with designed ARC controller for different vehicle masses.
324018.fig.0012
Figure 12: The yaw rate responses of the open-loop system, closed-loop systems with designed ARC controller for different vehicle masses.
324018.fig.0013
Figure 13: The tracking errors between real yaw rate and desired yaw rate of the open-loop system, closed-loop systems with designed ARC controller for different vehicle masses.
324018.fig.0014
Figure 14: Control inputs, .

The time histories of yaw rate for both open-loop systems and closed-loop systems with ARC controller in the case of different vehicle masses are shown in Figure 12, and Figure 13 is the corresponding responses of tracking errors between the real yaw rate and the desired yaw rate. From these figures, we can see that the proposed controller yields the least value of the maximum tracking error, compared with the open-loop system. Figure 14 shows the yaw moment, , for different cases (changing masses).

5. Concluding Remarks

In this paper, an adaptive robust control strategy has been proposed for vehicle lateral dynamic systems to improve vehicle handling and stability, where parameter uncertainties and external nonlinearities are considered in a unified framework. In order to manage two conflicting indexes, a compromise of the requirements is made, and the desired sideslip angle is replanned as a new reference signal, instead of the ideal “zero.” To the end, the designed controller can not only accomplish the required control purposes, but also effectively attenuate both the changes of vehicle mass and the variations of cornering stiffness. In addition, to overcome the problem of “explosion of complexity” caused by backstepping method in the traditional ARC design, the dynamic surface control (DSC) technique has been used to estimate the derivative of the virtual control. Finally, a nonlinear vehicle model is employed as the design example to illustrate the effectiveness of the proposed control law.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was supported in part by the self-planned task (no. SKLRS201308B) of the State Key Laboratory of Robotics and System (HIT), the Fundamental Research Funds for the Central Universities (no. HIT.NSRIF.2015032), China.

References

  1. A. R. W. Huang and C. Chen, “A low-cost driving simulator for full vehicle dynamics simulation,” IEEE Transactions on Vehicular Technology, vol. 52, no. 1, pp. 162–172, 2003. View at Publisher · View at Google Scholar · View at Scopus
  2. L. Cai, A. B. Rad, and W.-L. Chan, “An intelligent longitudinal controller for application in semiautonomous vehicles,” IEEE Transactions on Industrial Electronics, vol. 57, no. 4, pp. 1487–1497, 2010. View at Publisher · View at Google Scholar · View at Scopus
  3. D. Cao, X. Song, and M. Ahmadian, “Editors' perspectives: road vehicle suspension design, dynamics, and control,” Vehicle System Dynamics, vol. 49, no. 1-2, pp. 3–28, 2011. View at Publisher · View at Google Scholar · View at Scopus
  4. F. U. Rehman and M. M. Ahmed, “Steering control algorithm for a class of wheeled mobile robots,” IET Control Theory and Applications, vol. 1, no. 4, pp. 915–924, 2007. View at Publisher · View at Google Scholar · View at Scopus
  5. H. Xiao, W. Chen, H. Zhou, and J. W. Zu, “Integrated control of active suspension system and electronic stability programme using hierarchical control strategy: theory and experiment,” Vehicle System Dynamics, vol. 49, no. 1-2, pp. 381–397, 2011. View at Publisher · View at Google Scholar · View at Scopus
  6. H. Chou and B. D'Andréa-Novel, “Global vehicle control using differential braking torques and active suspension forces,” Vehicle System Dynamics, vol. 43, no. 4, pp. 261–284, 2005. View at Publisher · View at Google Scholar · View at Scopus
  7. J.-H. She, X. Xin, and Y. Ohyama, “Estimation of equivalent input disturbance improves vehicular steering control,” IEEE Transactions on Vehicular Technology, vol. 56, no. 6, pp. 3722–3731, 2007. View at Publisher · View at Google Scholar · View at Scopus
  8. J. Song and W. S. Che, “Comparison between braking and steering yaw moment controllers considering ABS control aspects,” Mechatronics, vol. 19, no. 7, pp. 1126–1133, 2009. View at Publisher · View at Google Scholar · View at Scopus
  9. M. Bertoluzzo and G. Buja, “Development of electric propulsion systems for light electric vehicles,” IEEE Transactions on Industrial Informatics, vol. 7, no. 3, pp. 428–435, 2011. View at Publisher · View at Google Scholar · View at Scopus
  10. P. Falcone, F. Borrelli, J. Asgari, H. E. Tseng, and D. Hrovat, “Predictive active steering control for autonomous vehicle systems,” IEEE Transactions on Control Systems Technology, vol. 15, no. 3, pp. 566–580, 2007. View at Publisher · View at Google Scholar · View at Scopus
  11. S. Han and K. Huh, “Monitoring system design for lateral vehicle motion,” IEEE Transactions on Vehicular Technology, vol. 60, no. 4, pp. 1394–1403, 2011. View at Publisher · View at Google Scholar · View at Scopus
  12. W. Cho, J. Yoon, J. Kim, J. Hur, and K. Yi, “An investigation into unified chassis control scheme for optimised vehicle stability and manoeuvrability,” Vehicle System Dynamics, vol. 46, no. 1, pp. 87–105, 2008. View at Publisher · View at Google Scholar · View at Scopus
  13. C. March and T. Shim, “Integrated control of suspension and front steering to enhance vehicle handling,” Journal of Automobile Engineering: Proceedings of the Institution of Mechanical Engineers Part D, vol. 221, no. 4, pp. 377–391, 2007. View at Publisher · View at Google Scholar · View at Scopus
  14. S.-B. Lu, Y.-N. Li, S.-B. Choi, L. Zheng, and M.-S. Seong, “Integrated control on MR vehicle suspension system associated with braking and steering control,” Vehicle System Dynamics, vol. 49, no. 1-2, pp. 361–380, 2011. View at Publisher · View at Google Scholar · View at Scopus
  15. Q. Zhou, P. Shi, J. Lu, and S. Xu, “Adaptive output-feedback fuzzy tracking control for a class of nonlinear systems,” IEEE Transactions on Fuzzy Systems, vol. 19, no. 5, pp. 972–982, 2011. View at Publisher · View at Google Scholar · View at Scopus
  16. H. Du, N. Zhang, and G. Dong, “Stabilizing vehicle lateral dynamics with considerations of parameter uncertainties and control saturation through robust yaw control,” IEEE Transactions on Vehicular Technology, vol. 59, no. 5, pp. 2593–2597, 2010. View at Publisher · View at Google Scholar · View at Scopus
  17. E. Gartley and D. M. Bevly, “Online estimation of implement dynamics for adaptive steering control of farm tractors,” IEEE/ASME Transactions on Mechatronics, vol. 13, no. 4, pp. 429–440, 2008. View at Publisher · View at Google Scholar · View at Scopus
  18. H. Lee and M. Tomizuka, “Adaptive vehicle traction force control for Intelligent Vehicle Highway Systems (IVHSs),” IEEE Transactions on Industrial Electronics, vol. 50, no. 1, pp. 37–47, 2003. View at Publisher · View at Google Scholar · View at Scopus
  19. J. Ahmadi, A. K. Sedigh, and M. Kabganian, “Adaptive vehicle lateral-plane motion control using optimal tire friction forces with saturation limits consideration,” IEEE Transactions on Vehicular Technology, vol. 58, no. 8, pp. 4098–4107, 2009. View at Publisher · View at Google Scholar · View at Scopus
  20. N. Ding and S. Taheri, “An adaptive integrated algorithm for active front steering and direct yaw moment control based on direct Lyapunov method,” Vehicle System Dynamics, vol. 48, no. 10, pp. 1193–1213, 2010. View at Publisher · View at Google Scholar · View at Scopus
  21. L. Xu and H. E. Tseng, “Robust model-based fault detection for a roll stability control system,” IEEE Transactions on Control Systems Technology, vol. 15, no. 3, pp. 519–528, 2007. View at Publisher · View at Google Scholar · View at Scopus
  22. Z. He and X. Ji, “Nonlinear robust control of integrated vehicle dynamics,” Vehicle System Dynamics, vol. 50, no. 2, pp. 247–280, 2012. View at Publisher · View at Google Scholar · View at Scopus
  23. H. Karimi, M. Zapateiro, and N. Luo, “An LMI approach to H control of vehicle engine-body vibration systems with time-varying actuator delay,” Journal of Systems and Control Engineering: Proceedings of the Institution of Mechanical Engineers Part I, vol. 222, no. 8, pp. 883–899, 2008. View at Google Scholar
  24. H. Li, X. Jing, and H. Karimi, “Output-feedback based H control for active suspension systems with control delay,” IEEE Transactions on Industrial Electronics, vol. 61, no. 1, pp. 436–446, 2014. View at Publisher · View at Google Scholar
  25. W. Sun, Y. Zhao, J. Li, L. Zhang, and H. Gao, “Active suspension control with frequency band constraints and actuator input delay,” IEEE Transactions on Industrial Electronics, vol. 59, no. 1, pp. 530–537, 2012. View at Publisher · View at Google Scholar · View at Scopus
  26. Z. Wang, Y. Liu, and X. Liu, “H filtering for uncertain stochastic time-delay systems with sector-bounded nonlinearities,” Automatica, vol. 44, no. 5, pp. 1268–1277, 2008. View at Publisher · View at Google Scholar · View at Scopus
  27. H. R. Karimi, “Optimal vibration control of vehicle engine-body system using Haar functions,” International Journal of Control, Automation and Systems, vol. 4, no. 6, pp. 714–724, 2006. View at Google Scholar · View at Scopus
  28. H. Li, H. Liu, H. Gao, and P. Shi, “Reliable fuzzy control for active suspension systems with actuator delay and fault,” IEEE Transactions on Fuzzy Systems, vol. 20, no. 2, pp. 342–357, 2012. View at Publisher · View at Google Scholar · View at Scopus
  29. J. Cao, P. Li, and H. Liu, “An interval fuzzy controller for vehicle active suspension systems,” IEEE Transactions on Intelligent Transportation Systems, vol. 11, no. 4, pp. 885–895, 2010. View at Publisher · View at Google Scholar · View at Scopus
  30. H. Li, J. Yu, C. Hilton, and H. Liu, “Adaptive sliding mode control for nonlinear active suspension vehicle systems using T-S fuzzy approach,” IEEE Transactions on Industrial Electronics, vol. 60, no. 8, pp. 3328–3338, 2013. View at Publisher · View at Google Scholar
  31. O. Kaynak and A. Denker, “Discrete-time sliding mode control in the presence of system uncertainty,” International Journal of Control, vol. 57, no. 5, pp. 1177–1189, 1993. View at Publisher · View at Google Scholar
  32. X. Yu, M. O. Efe, and O. Kaynak, “A general backpropagation algorithm for feedforward neural networks learning,” IEEE Transactions on Neural Networks, vol. 13, no. 1, pp. 251–254, 2002. View at Publisher · View at Google Scholar · View at Scopus
  33. Y. Tang, H. Gao, W. Zou, and J. Kurths, “Distributed synchronization in networks of agent systems with nonlinearities and random switchings,” IEEE Transactions on Cybernetics, vol. 43, no. 1, pp. 358–370, 2013. View at Publisher · View at Google Scholar
  34. Y. Tang and W. Wong, “Distributed synchronization of coupled neural networks via randomly occurring control,” IEEE Transactions on Neural Networks and Learning Systems, vol. 24, no. 3, pp. 435–447, 2013. View at Publisher · View at Google Scholar
  35. Q. Zhou, P. Shi, H. Liu, and S. Xu, “Neural network based decentralized adaptive output-feedback control for large-scale stochastic nonlinear systems,” IEEE Transactions on Systems, Man, and Cybernetics B, vol. 42, no. 6, pp. 1608–1619, 2012. View at Publisher · View at Google Scholar
  36. Q. Zhou, P. Shi, S. Xu, and H. Li, “Observer-based adaptive neural network control for nonlinear stochastic systems with time-delay,” IEEE Transactions on Neural Networks and Learning Systems, vol. 24, no. 1, pp. 71–80, 2013. View at Publisher · View at Google Scholar
  37. M. Zapateiro, N. Luo, H. R. Karimi, and J. Vehí, “Vibration control of a class of semiactive suspension system using neural network and backstepping techniques,” Mechanical Systems and Signal Processing, vol. 23, no. 6, pp. 1946–1953, 2009. View at Publisher · View at Google Scholar · View at Scopus
  38. M. Zapateiro, F. Pozo, H. R. Karimi, and N. Luo, “Semiactive control methodologies for suspension control with magnetorheological dampers,” IEEE/ASME Transactions on Mechatronics, vol. 17, no. 2, pp. 370–380, 2012. View at Publisher · View at Google Scholar · View at Scopus
  39. L. Zuo, J.-J. E. Slotine, and S. A. Nayfeh, “Model reaching adaptive control for vibration isolation,” IEEE Transactions on Control Systems Technology, vol. 13, no. 4, pp. 611–617, 2005. View at Publisher · View at Google Scholar · View at Scopus
  40. S. Yin, H. Luo, and S. Ding, “Real-time implementation of fault-tolerant control systems with performance optimization,” IEEE Transactions on Industrial Electronics, vol. 61, no. 5, pp. 2402–2411, 2014. View at Publisher · View at Google Scholar
  41. S. Yin, S. Ding, A. Haghani, H. Hao, and P. Zhang, “A comparison study of basic data-driven fault diagnosis and process monitoring methods on the benchmark Tennessee Eastman process,” Journal of Process Control, vol. 22, no. 9, pp. 1567–1581, 2012. View at Publisher · View at Google Scholar
  42. B. Yao, F. Bu, J. Reedy, and G. T.-C. Chiu, “Adaptive robust motion control of single-rod hydraulic actuators: theory and experiments,” IEEE/ASME Transactions on Mechatronics, vol. 5, no. 1, pp. 79–91, 2000. View at Publisher · View at Google Scholar · View at Scopus
  43. J. Yao, Z. Jiao, and D. Ma, “Adaptive robust control of DC motors with extended state observer,” IEEE Transactions on Industrial Electronics, vol. 61, no. 7, pp. 3630–3637, 2014. View at Publisher · View at Google Scholar
  44. J. Yao, Z. Jiao, D. Ma, and L. Yan, “High-accuracy tracking control of hydraulic rotary actuators with modeling uncertainties,” IEEE/ASME Transactions on Mechatronics, no. 99, pp. 1–9, 2013. View at Publisher · View at Google Scholar
  45. H. Liu, “Exploring human hand capabilities into embedded multifingered object manipulation,” IEEE Transactions on Industrial Informatics, vol. 7, no. 3, pp. 389–398, 2011. View at Publisher · View at Google Scholar · View at Scopus