Recursive Least Squares Algorithm for a Nonlinear Additive System with Time Delay

doi:10.1007/s12204-016-1707-2

Abstract

Abstract: This paper proposes a recursive least squares algorithm for a nonlinear additive system with time delay. By the Weierstrass approximation theorem and the key term separation principle, the model can be simplified as an identification model. Based on the identification model, a recursive least squares identification algorithm is used to estimate all the unknown parameters of the time-delayed additive system. An example is provided to show the effectiveness of the proposed algorithm.

Key words: parameter estimation| recursive least square algorithm| Weierstrass approximation theorem| key term separation principle| additive system

摘要： This paper proposes a recursive least squares algorithm for a nonlinear additive system with time delay. By the Weierstrass approximation theorem and the key term separation principle, the model can be simplified as an identification model. Based on the identification model, a recursive least squares identification algorithm is used to estimate all the unknown parameters of the time-delayed additive system. An example is provided to show the effectiveness of the proposed algorithm.

关键词: parameter estimation| recursive least square algorithm| Weierstrass approximation theorem| key term separation principle| additive system

CLC Number:

TP 273

CHEN Jing1,2* (陈晶), WANG Xiuping2 (王秀平). Recursive Least Squares Algorithm for a Nonlinear Additive System with Time Delay[J]. Journal of shanghai Jiaotong University (Science), 2016, 21(2): 159-163.

References 25

[1]	BAI E W. Identification of nonlinear additive FIR systems[J]. Automatica, 2005, 41(7): 1247-1253.
[2]	BAI E W, CHAN K S. Identification of an additive nonlinear system and its applications in generalized Hammerstein models [J]. Automatica, 2008, 44(2):430-436.
[3]	GU Y, DING F, LI J H. States based iterative parameter estimation for a state space model with multistate delays using decomposition [J]. Signal Processing,2015, 106: 294-300.
[4]	WANG D Q, SHAN T, DING R. Data filtering based stochastic gradient algorithms for multivariable CARAR-like systems [J]. Mathematical Modelling and Analysis, 2013, 18(3): 374-385.
[5]	WANG X H, DING F. Performance analysis of the recursive parameter estimation algorithms for multivariable Box-Jenkins systems [J]. Journal of the Franklin Institute, 2014, 351(10): 4749-4764.
[6]	WANG C, TANG T. Recursive least squares estimation algorithm applied to a class of linear-inparameters output error moving average systems [J].Applied Mathematics Letters, 2014, 29: 36-41.
[7]	DING F, DENG K P, LIU X M. Decomposition based Newton iterative identification method for a Hammerstein nonlinear FIR system with ARMA noise [J].Circuits, Systems and Signal Processing, 2014, 33(9):2881-2893.
[8]	LIU Y, BAI E W. Iterative identification of Hammerstein systems [J]. Automatica, 2007, 43(2): 346-354.
[9]	DING F, CHEN T W. Performance analysis of multiinnovation gradient type identification methods [J].Automatica, 2007, 43(1): 1-14.
[10]	GOODWIN G C, SIN K S. Adaptive filtering prediction and control [M]. Englewood Cliffs, NJ: Prentice-Hall, 1984.
[11]	SHI P, LUAN X L, LIU F. H∞ filtering for discretetime systems with stochastic incomplete measurement and mixed delays [J]. IEEE Transactions on Industrial Electronics, 2012, 59(6): 2732-2739.
[12]	LUAN X L, SHI P, LIU F. Stabilization of networked control systems with random delays [J]. IEEE Transactions on Industrial Electronics, 2011, 58(9): 4323-4330.
[13]	SHI Y, YU B. Robust mixed H2/H∞ control of networked control systems with random time delays in both forward and backward communication links [J].Automatica, 2011, 47(4): 754-760.
[14]	SONG Y, WANG J C, SHI Y H, et al. H∞ control of networked control systems with time delay and packet disordering [J]. Journal of the Franklin Institute, 2013,350(6): 1596-1616.
[15]	TIAN E G, YUE D, ZHANG Y J. On improved delaydependent robust H∞ control for systems with interval time-varying delay [J]. Journal of the Franklin Institute,2011, 348(4): 555-567.
[16]	GU Y, DING R. A least squares numerical algorithm for state space model with multi-state delays [J]. Applied Mathematics Letters, 2013, 26(7): 748-753.
[17]	GU Y, DING F, LI J H. State filtering and parameter estimation for linear systems with d-step state-delay[J]. IET Signal Processing, 2014, 8(6): 639-646.
[18]	BAI EW. Identification of linear systems with hard input nonlinearities of known structure [J]. Automatica,2002, 38(5): 853-860.
[19]	CHEN J, WANG X P, DING R F. Gradient based estimation algorithm for Hammerstein systems with saturation and dead-zone nonlinearities [J]. Applied Mathematical Modelling, 2012, 36(1): 238-243.
[20]	V¨OR¨OS J. Modeling and identification of systems with backlash [J]. Automatica, 2010, 46(2): 369-374.
[21]	ROCHDI Y, GIRI F, GNING J B, et al. Identification of block-oriented systems in the presence of nonparametric input nonlinearities of switch and backlash types [J]. Automatica, 2010, 46(5): 864-867.
[22]	V¨OR¨OS J. Identification of Hammerstein systems with time-varying piecewise linear characteristics [J]. IEEE Transactions on Circuits and Systems, 2005, 52 (12):865-869.
[23]	WANG D Q, CHU Y Y, YANG G W, et al. Auxiliary model-based recursive generalized least squares parameter estimation for Hammerstein OEAR systems [J].Mathematical and Computer Modelling, 2010, 52(1/2):309-317.
[24]	V¨OR¨OS J. Iterative algorithm for parameter identification of Hammerstein systems with two-segment nonlinearities[J]. IEEE Transactions on Automatic control,1999, 44(11): 2145-2149.
[25]	V¨OR¨OS J. Parameter identification of discontinuous Hammerstein systems [J]. Automatica, 1997, 33(6):1141-1146.