Research Papers

Robust Stochastic Sampled-Data H Control for a Class of Mechanical Systems With Uncertainties

[+] Author and Article Information
S. Dharani

Department of Mathematics,
Bharathiar University,
Coimbatore 641046, Tamil Nadu, India
e-mail: sdharanimails@gmail.com

R. Rakkiyappan

Department of Mathematics,
Bharathiar University,
Coimbatore 641046, Tamil Nadu, India
e-mail: rakkigru@gmail.com

Jinde Cao

Department of Mathematics and
Research Center for Complex Systems and
Network Sciences,
Southeast University,
Nanjing 210096, Jiangsu, China;
Department of Mathematics,
Faculty of Science,
King Abdulaziz University,
Jeddah 21589, Saudi Arabia
e-mail: jdcao@seu.edu.cn

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received August 15, 2014; final manuscript received June 5, 2015; published online July 21, 2015. Assoc. Editor: Umesh Vaidya.

J. Dyn. Sys., Meas., Control 137(10), 101008 (Jul 21, 2015) Paper No: DS-14-1333; doi: 10.1115/1.4030800 History: Received August 15, 2014

This paper considers a class of mechanical systems with uncertainties appearing in all the mass, damping, and stiffness matrices. Two cases, linear fractional and randomly occurring uncertainty formulations, are considered. Since sampled-data controllers have an advantage of implementing with microcontroller or digital computer to lower the implementation cost and time, a robust stochastic sampled-data controller is considered with m sampling intervals whose occurrence probabilities are given constants and satisfy Bernoulli distribution. A discontinuous type Lyapunov functional based on the extended Wirtinger's inequality is constructed with triple integral terms and sufficient conditions that promises the robust mean square asymptotic stability of the concerned system are derived in terms of linear matrix inequalities (LMIs). In an aim to reduce the conservatism, a newly introduced concept called the second-order reciprocally convex approach is employed in deriving the bound for some cross terms that arise while maneuvering the derivative of Lyapunov functional. The obtained LMIs can be easily solved through any of the standard available software. Finally, numerical examples are given to verify the effectiveness of the proposed theoretical results.

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Campion, G. , d'Andrea-Novel, B. , and Bastin, G. , 1991, “Controllability and State Feedback Stabilizability of Nonholonomic Mechanical Systems,” International Workshop on Nonlinear and Adaptive Control: Issues in Robotics, Grenoble, France, pp. 106–124.
Potsaid, B. , and Wen, J. T. , 2002, “Optimal Mechanical Design of a Rotary Inverted Pendulum,” IEEE/RSJ International Conference on IROS, EPFL, Laussanne, Switzerland, pp. 2079–2084.
Yao, B. , and Xu, L. , 2002, “Adaptive Robust Motion Control of Linear Motors for Precision Manufacturing,” Mechatronics, 12(4), pp. 595–616. [CrossRef]
Tang, K. Z. , Huang, S. N. , Tan, K. K. , and Lee, T. H. , 2004, “Combined PID and Adaptive Nonlinear Control for Servo Mechanical Systems,” Mechatronics, 14(6), pp. 701–714. [CrossRef]
Xia, Y. , Shi, P. , Liu, G. , and Rees, D. , 2005, “Robust Mixed H 2/H State Feedback Control for Continuous-Time Descriptor Systems With Parameter Uncertainties,” Circuits, Syst., Signal Process., 24(4), pp. 431–443. [CrossRef]
Kar, I. N. , Miyakura, T. , and Seto, K. , 2000, “Bending and Torsional Vibration Control of a Flexible Plate Structure Using H 1-Based Robust Control Law,” IEEE Trans. Control Syst. Technol., 8(3), pp. 545–553. [CrossRef]
Shi, P. , 1998, “Filtering on Sampled-Data Systems With Parametric Uncertainty,” IEEE Trans. Autom. Control, 43(7), pp. 1022–1027. [CrossRef]
Li, F. , Shi, P. , and Lim, C.-C. , 2015, “State Estimation and Sliding Mode Control for Semi-Markovian Jump Systems With Mismatched Uncertainties,” Automatica, 51, pp. 385–393. [CrossRef]
Li, T. , Guo, L. , and Lin, C. , 2007, “A New Criterion of Delay Dependent Stability for Uncertain Time-Delay Systems,” IET Control Theory Appl., 1(3), pp. 611–616. [CrossRef]
Li, T. , Guo, L. , and Sun, C. , 2007, “Robust Stability for Neural Networks With Time-Varying Delays and Linear Fractional Uncertainties,” Neurocomputing, 71(1–3), pp. 421–427. [CrossRef]
Balasubramaniam, P. , and Lakshmanan, S. , 2011, “Delay-Interval-Dependent Robust-Stability Criteria for Neutral Stochastic Neural Networks With Polytopic and Linear Fractional Uncertainties,” Int. J. Comput. Math., 88(10), pp. 2001–2015. [CrossRef]
Balasubramaniam, P. , Lakshmanan, S. , and Rakkiyappan, R. , 2012, “LMI Optimization Problem of Delay-Dependent Robust Stability Criteria for Stochastic Systems With Polytopic and Linear Fractional Uncertainties,” Int. J. Appl. Math. Comput. Sci., 22(2), pp. 339–351. [CrossRef]
Zames, G. , 1981, “Feedback and Optimal Sensitivity Model Reference Transformations, Multiplicative Seminorms and Approximate Inverses,” IEEE Trans. Autom. Control, 26(2), pp. 301–320. [CrossRef]
Huang, J. , and Shi, Y. , 2013, “ H State Feedback Control for Semi-Markov Jump Linear Systems With Time-Varying Delays,” ASME J. Dyn. Syst. Meas. Control, 135(4), p. 041012. [CrossRef]
Wang, C. , and Shen, Y. , 2012, “Robust H Control for Stochastic Systems With Nonlinearity, Uncertainty and Time-Varying Delay,” Comput. Math. Appl., 63(5), pp. 985–998. [CrossRef]
Yazici, H. , Kucukdemiral, I. B. , Parlakci, M. N. A. , and Guclu, R. , 2012, “Robust Delay-Dependent H Control for Uncertain Structural Systems With Actuator Delay,” ASME J. Dyn. Syst. Meas. Control, 134(3), p. 031013. [CrossRef]
Yang, X. , Gao, H. , and Shi, P. , 2010, “Robust H Control for a Class of Uncertain Mechanical Systems,” Int. J. Control, 83(7), pp. 1303–1324. [CrossRef]
Du, H. , Lam, J. , and Sze, K. , 2005, “ H Disturbance Attenuation for Uncertain Mechanical Systems With Input Delay,” Trans. Inst. Meas. Control, 27(1), pp. 37–52. [CrossRef]
Lei, J. , 2013, “Optimal Vibration Control for Uncertain Nonlinear Sampled-Data Systems With Actuator and Sensor Delays: Application to a Vehicle Suspension,” ASME J. Dyn. Syst. Meas. Control, 135(2), p. 021021. [CrossRef]
Lam, H. , 2012, “Stabilization of Nonlinear Systems Using Sampled-Data Output-Feedback Fuzzy Controller Based on Polynomial-Fuzzy-Model-Based Control Approach,” IEEE Trans. Syst. Man Cybern. Part B Cybern., 42(1), pp. 258–267. [CrossRef]
Pertew, A. W. , Marquez, H. J. , and Zhao, Q. , 2009, “Sampled-Data Stabilization of a Class of Nonlinear Systems With Applications in Robotics,” ASME J. Dyn. Syst. Meas. Control, 131(2), p. 021008. [CrossRef]
Astrom, K. , and Wittenmark, B. , 1989, Adaptive Control, Addison-Wesley, Reading, MA.
Mikheev, Y. V. , Sobolev, V. A. , and Fridman, E. , 1988, “Asymptotic Analysis of Digital Control Systems,” Autom. Remote Control, 49(9), pp. 1175–1180.
Tahara, S. , Fujii, T. , and Yokoyama, T. , 2007, “Variable Sampling Quasi Multirate Deadbeat Control Method for Single Phase PWM Inverter in Low Carrier Frequency,” Power Conversion Conference, Nagoya, Japan, Apr. 2–5, pp. 804–809.
Hu, B. , and Michel, A. N. , 2000, “Stability Analysis of Digital Feedback Control Systems With Time-Varying Sampling Periods,” Automatica, 36(6), pp. 897–905. [CrossRef]
Shen, B. , Wang, Z. , and Liu, X. , 2012, “Sampled-Data Synchronization Control of Dynamical Networks With Stochastic Sampling,” IEEE Trans. Autom. Control, 57(10), pp. 2644–2650. [CrossRef]
Gao, H. , Wu, J. , and Shi, P. , 2009, “Robust Sampled-Data H Control With Stochastic Sampling,” Automatica, 45(7), pp. 1729–1736. [CrossRef]
Lee, T. H. , Park, J. H. , Lee, S. M. , and Kwon, O. M. , 2013, “Robust Synchronization of Chaotic Systems With Randomly Occurring Uncertainties Via Stochastic Sampled-Data Control,” Int. J. Control, 86(1), pp. 107–119. [CrossRef]
Lee, T. H. , Park, J. H. , Kwon, O. M. , and Lee, S. M. , 2013, “Stochastic Sampled-Data Control for State Estimation of Time-Varying Delayed Neural Networks,” Neural Network, 46, pp. 99–108. [CrossRef]
Bamieh, B. , and Pearson, J. , 1992, “A General Framework for Linear Periodic Systems With Applications to H Sampled-Data Control,” IEEE Trans. Autom. Control, 37(4), pp. 418–435. [CrossRef]
Sivashankar, N. , and Khargonekar, P. , 1994, “Characterization of the L 2-Induced Norm for Linear Systems With Jumps With Applications to Sampled-Data Systems,” SIAM J. Control Optim., 32(4), pp. 1128–1150. [CrossRef]
Li, F. , Shi, P. , Wu, L. , and Zhang, X. , 2014, “Fuzzy-Model-Based-Stability and Nonfragile Control for Discrete-Time Descriptor Systems With Multiple Delays,” IEEE Trans. Fuzzy Syst., 22(4), pp. 1019–1025. [CrossRef]
Fridman, E. , Shaked, U. , and Suplin, V. , 2005, “Input/Output Delay Approach to Robust Sampled-Data H Control,” Syst. Control Lett., 54(3), pp. 271–282. [CrossRef]
Fridman, E. , and Am, N. B. , 2013, “Sampled-Data Distributed H Control of Transport Reaction Systems,” SIAM J. Control Optim., 51(2), pp. 1500–1527. [CrossRef]
Gao, H. , Sun, W. , and Shi, P. , 2010, “Robust Sampled-Data H Control for Vehicle Active Suspension Systems,” IEEE Trans. Control Syst. Technol., 18(1), pp. 238–245. [CrossRef]
Weng, Y. , and Chao, Z. , 2014, “Robust Sampled-Data H Output Feedback Control of Active Suspension System,” Int. J. Innovative Comput. Inf. Control, 10(1), pp. 281–292.
Lien, C. H. , Chen, J. D. , Yu, K. W. , and Chung, L. Y. , 2012, “Robust Delay-Dependent H Control for Uncertain Switched Time-Delay Systems Via Sampled-Data State Feedback Input,” Comput. Math. Appl., 64(5), pp. 1187–1196. [CrossRef]
Ren, F. , and Cao, J. , 2006, “Novel α – Stability Criterion of Linear Systems With Multiple Time Delays,” Appl. Math. Comput., 181(1), pp. 282–290. [CrossRef]
Liu, F. , He, F. , and Yao, Y. , 2008, “Linear Matrix Inequality-Based Robust H Control of Sampled-Data Systems With Parametric Uncertainties,” IET Control Theory Appl., 2(4), pp. 253–260. [CrossRef]
Park, P. G. , Ko, J. W. , and Jeong, C. , 2011, “Reciprocally Convex Approach to Stability of Systems With Time-Varying Delays,” Automatica, 47(1), pp. 235–238. [CrossRef]
Liu, J. , and Zhang, J. , 2012, “Note on Stability of Discrete-Time Time-Varying Delay Systems,” IET Control Theory Appl., 6(2), pp. 335–339. [CrossRef]
Feng, Z. , and Lam, J. , 2012, “Integral Partitioning Approach to Robust Stabilization for Uncertain Distributed Time-Delay Systems,” Int. J. Robust Nonlinear Control, 22(6), pp. 676–689. [CrossRef]
Lee, W. , and Park, P. , 2014, “Second-Order Reciprocally Convex Approach to Stability of Systems With Interval Time-Varying Delays,” Appl. Math. Comput., 229, pp. 245–253. [CrossRef]
Sakthivel, R. , Arunkumar, A. , and Mathiyalagan, K. , 2015, “Robust Sampled-Data H Control for Mechanical Systems,” Complexity, 20(4), pp. 19–29. [CrossRef]


Grahic Jump Location
Fig. 1

State trajectories of displacement and velocity of controlled nominal system

Grahic Jump Location
Fig. 2

State trajectories of displacement and velocity of open-loop nominal system

Grahic Jump Location
Fig. 6

State trajectories of displacement and velocity of uncontrolled mechanical system with randomly occurring uncertainties

Grahic Jump Location
Fig. 7

Stochastic parameter h of Example 1

Grahic Jump Location
Fig. 8

Stochastic parameter h of Example 2

Grahic Jump Location
Fig. 5

State trajectories of displacement and velocity of controlled mechanical system with randomly occurring uncertainties

Grahic Jump Location
Fig. 9

Time evolutions of δ(t) ; δ(t) switch from values 0 and 1 according to their expectations in Example 3

Grahic Jump Location
Fig. 3

State trajectories of displacement and velocity of closed-loop uncertain system

Grahic Jump Location
Fig. 4

State trajectories of displacement and velocity of uncontrolled uncertain system




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