Adaptive neural tracking control for a class of stochastic nonlinear systems
Huan-qing Wang
Institute of Complexity Science, Qingdao University, Qingdao 266071, Shandong, China
School of Mathematics and Physics, Bohai University, Jinzhou 121000, Liaoning, China
Search for more papers by this authorCorresponding Author
Bing Chen
Institute of Complexity Science, Qingdao University, Qingdao 266071, Shandong, China
Correspondence to: Bing Chen, Institute of Complexity Science, Qingdao University, Qingdao 266071, Shandong, China.
E-mail: [email protected]
Search for more papers by this authorChong Lin
Institute of Complexity Science, Qingdao University, Qingdao 266071, Shandong, China
Search for more papers by this authorHuan-qing Wang
Institute of Complexity Science, Qingdao University, Qingdao 266071, Shandong, China
School of Mathematics and Physics, Bohai University, Jinzhou 121000, Liaoning, China
Search for more papers by this authorCorresponding Author
Bing Chen
Institute of Complexity Science, Qingdao University, Qingdao 266071, Shandong, China
Correspondence to: Bing Chen, Institute of Complexity Science, Qingdao University, Qingdao 266071, Shandong, China.
E-mail: [email protected]
Search for more papers by this authorChong Lin
Institute of Complexity Science, Qingdao University, Qingdao 266071, Shandong, China
Search for more papers by this authorSUMMARY
This paper investigates the problem of adaptive neural control design for a class of single-input single-output strict-feedback stochastic nonlinear systems whose output is an known linear function. The radial basis function neural networks are used to approximate the nonlinearities, and adaptive backstepping technique is employed to construct controllers. It is shown that the proposed controller ensures that all signals of the closed-loop system remain bounded in probability, and the tracking error converges to an arbitrarily small neighborhood around the origin in the sense of mean quartic value. The salient property of the proposed scheme is that only one adaptive parameter is needed to be tuned online. So, the computational burden is considerably alleviated. Finally, two numerical examples are used to demonstrate the effectiveness of the proposed approach. Copyright © 2012 John Wiley & Sons, Ltd.
REFERENCES
- 1 Krstic M, Kanellakopoulos I, Kokotovic PV. Nonlinear and Adaptive Control Design. Wiley Interscience: New York, 1995.
- 2 Polycarpou MM. Stable adaptive neural control scheme for nonlinear systems. IEEE Transactions on Automatic Control 1996; 41(3): 447–451.
- 3 Zhang T, Ge SS, Hang CC. Stable adaptive control for a class of nonlinear systems using a modified Lyapunov function. IEEE Transactions on Automatic Control 2000; 45(1): 129–132.
- 4 Chen B, Liu XP, Liu KF, Lin C. Direct adaptive fuzzy control of nonlinear strict-feedback systems. Automatica 2009; 45(6): 1530–1535.
- 5 Chen B, Liu XP, Liu KF, Lin C. Fuzzy-approximation-based adaptive control of strict-feedback nonlinear systems with time delays. IEEE Transactions on Fuzzy Systems 2010; 18(5): 883–892.
- 6 Tong SC, Li YM. Observer-based fuzzy adaptive control for strict-feedback nonlinear systems. Fuzzy Sets and Systems 2009; 160(12): 1749–1764.
- 7 Tong SC, Li CY, Li YM. Fuzzy adaptive observer backstepping control for MIMO nonlinear systems. Fuzzy Sets and Systems 2009; 160(19): 2755–2775.
- 8 Liu YJ, Tong SC, Li YM. Adaptive neural network tracking control for a class of nonlinear systems. International Journal of Systems Science 2010; 41(2): 143–158.
- 9 Li TS, Tong SC, Feng G. A novel robust adaptive fuzzy tracking control for a class of nonlinear multi-input ∕ multi-output systems. IEEE Transactions on Fuzzy Systems 2010; 18(1): 150–160.
- 10 Tong SC, He XL, Zhang HG. A combined backstepping and small-gain approach to robust adaptive fuzzy output feedback control. IEEE Transactions on Fuzzy Systems 2009; 17(5): 1059–1069.
- 11 Pan Z, Basa T. Backstepping controller design for nonlinear stochastic systems under a risk-sensitive cost criterion. SIAM Journal on Control and Optimization 1999; 37(3): 957–995.
- 12 Liu YG, Pan ZG, Shi SJ. Output feedback control design for strict-feedback stochastic nonlinear systems under a risk-sensitive cost. IEEE Transactions on Automatic Control 2003; 48(3): 509–513.
- 13 Pan ZG, Ezal K, Krener A, Kokotovic PV. Backstepping design with local optimality matching. IEEE Transactions on Automatic Control 2001; 46(7): 1014–1027.
- 14 Liu YG, Zhang JF. Practical output feedback risk-sensitive control for stochastic nonlinear systems under stable zero-dynamics. SIAM Journal on Control and Optimization 2006; 45(3): 885–926.
- 15 Wu ZJ, Xie XJ, Zhang SY. Adaptive backstepping controller design using stochastic small-gain theorem. Automatica 2007; 43(4): 608–620.
- 16 Deng H, Krstić M. Stochastic nonlinear stabilization, Part I: a backstepping design. Systems & Control Letters 1997; 32(3): 143–150.
- 17 Deng H, Krstić M. Stochastic nonlinear stabilization, Part II: inverse optimality. Systems & Control Letters 1997; 32(3): 151–159.
- 18 Deng H, Krstić M, Willians RJ. Stabilization of stochastic nonlinear systems driven by noise of unknown covariance. IEEE Transactions on Automatic Control 2001; 46(8): 1237–1253.
- 19 Liu SJ, Zhang JF, Jiang ZP. Decentralized adaptive output-feedback stabilization for large-scale stochastic nonlinear systems. Automatica 2007; 43(2): 238–251.
- 20 Xie SL, Xie LH. Decentralized stabilization of a class of interconnected stochastic nonlinear systems. IEEE Transactions on Automatic Control 2000; 45(1): 132–137.
- 21 Ji HB, Xi HS. Adaptive output-feedback tracking of stochastic nonlinear systems. IEEE Transactions on Automatic Control 2006; 51(2): 355–360.
- 22 Xie XJ, Tian J. Adaptive state-feedback stabilization of high-order stochastic systems with nonlinear parameterization. Automatica 2009; 45(1): 126–133.
- 23 Xie XJ, Tian J. State-feedback stabilization for high-order stochastic nonlinear systems with stochastic inverse dynamics. International Journal of Robust and Nonlinear Control 2007; 17(14): 1343–1362.
- 24 Liu SJ, Ge SS, Zhang JF. Adaptive output-feedback control for a class of uncertain stochastic non-linear systems with time delays. International Journal of Control 2008; 81(8): 1210–1220.
- 25 Wang M, Zhang SY, Chen B, Luo F. Direct adaptive neural control for stabilization of nonlinear time-delay systems. Science in China Series F: Information Sciences 2010; 53(4): 800–812.
- 26 Wang M, Ge SS. Approximation-based adaptive tracking control of pure-feedback nonlinear systems with multiple unknown time-varying delays. Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference, Shanghai, China, 2009; 841–846.
- 27 Wang C, Hill DJ, Ge SS, Chen GR. An ISS-modular approach for adaptive neural control of pure-feedback systems. Automatica 2006; 42(5): 723–731.
- 28 Psillakis HE, Alexandridis AT. Adaptive neural tracking for a class of SISO uncertain and stochastic nonlinear systems. In Proceedings of IEEE Conference on Decision Control-European Control Conference, Seville, Spain, 2005; 7822–7827.
- 29 Psillakis HE, Alexandridis AT. NN-based adaptive tracking control of uncertain nonlinear systems disturbed by unknown covariance noise. IEEE Transactions on Neural Networks 2007; 18(6): 1830–1835.
- 30 Li J, Chen WS, Li JM. Adaptive NN output-feedback decentralized stabilization for a class of large-scale stochastic nonlinear strict-feedback systems. International Journal of Robust and Nonlinear Control 2011; 21(3): 452–472.
- 31 Wang YC, Zhang HG, Wang YZ. Fuzzy adaptive control of stochastic nonlinear systems with unknown virtual control gain function. Acta Automatica Sinica 2006; 32(2): 170–178.
- 32 Yu ZX, Du HB. Neural-network-based bounded adaptive stabilization for uncertain stochastic nonlinear systems with time-delay. Journal of Control Theory and Applications 2010; 27(7): 855–860.
- 33 Yu ZX, Du HB. Adaptive neural tracking control for stochastic nonlinear systems with time-varying delay. Journal of Control Theory and Applications 2011; 28(2): 1808–1812.
- 34
Yang YS,
Zhou C,
Ren JS. Model reference adaptive robust fuzzy control for ship steering autopilot with uncertain nonlinear systems. Applied Soft Computing 2003; 3(4): 305–316.
10.1016/j.asoc.2003.05.001 Google Scholar
- 35 Yang YS, Feng G, Ren JS. A combined backstepping and small-gain approach to robust adaptive fuzzy control for strict-feedback nonlinear systems. IEEE Transactions on Systems, Man, and Cybernetics-Part A: Systems and Humans 2004; 34(3): 406–420.
- 36
Khas'minskii RZ. Stochastic Stability of Differential Equations. Kluwer Academic Publishers: Norwell, MA, 1980.
10.1007/978-94-009-9121-7 Google Scholar
- 37 Sanner RM, Slotine JE. Gaussian networks for direct adaptive control. IEEE Transactions on Neural Network 1992; 3(6): 837–863.
- 38 Kurdila AJ, Narcowich FJ, Ward JD. Persistency of excitation in identification using radial basis function approximants. SIAM Journal of Control and Optimization 1995; 33(2): 625–642.
- 39 Ge SS, Wang J. Robust adaptive neural control for a class of perturbed strict feedback nonlinear systems. IEEE Transactions on Neural Networks 2002; 13(6): 1409–1419.
