×

Least squares algorithm for an input nonlinear system with a dynamic subspace state space model. (English) Zbl 1281.93050

Summary: For a Hammerstein input nonlinear system with a subspace state space linear element, this paper transforms the system into a bilinear identification model by using the property of the shift operator to the state space model and presents a recursive and an iterative least squares algorithms to generate parameter estimates and state estimates by using the hierarchical identification principle and by replacing the unknown state variables with their estimates. The proposed approaches are computationally more efficient than the over-parameterization model based least squares method.

MSC:

93C10 Nonlinear systems in control theory
93E24 Least squares and related methods for stochastic control systems
93E12 Identification in stochastic control theory
93E10 Estimation and detection in stochastic control theory
93C25 Control/observation systems in abstract spaces
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Ding, F.: System Identification—New Theory and Methods. Science Press, Beijing (2013)
[2] Rashid, M.T.; Frasca, M.; Ali, A.A.; Ali, R.S.; Fortuna, L.; Xibilia, M.G., Nonlinear model identification for artemia population motion, Nonlinear Dyn., 69, 2237-2243, (2012)
[3] Farjoud, A.; Ahmadian, M.; Craft, M.; Burke, W., Nonlinear modeling and experimental characterization of hydraulic dampers: effects of shim stack and orifice parameters on damper performance, Nonlinear Dyn., 67, 1437-1456, (2012)
[4] Gaite, J., Nonlinear analysis of spacecraft thermal models, Nonlinear Dyn., 65, 283-300, (2011)
[5] Shams, S.; Sadr, M.H.; Haddadpour, H., An efficient method for nonlinear aeroelasticity of slender wings, Nonlinear Dyn., 67, 659-681, (2012) · Zbl 1356.74064
[6] Ding, F.; Liu, X.G.; Chu, J., Gradient-based and least-squares-based iterative algorithms for Hammerstein systems using the hierarchical identification principle, IET Control Theory Appl., 7, 176-184, (2013)
[7] Smith, J.G.; Kamat, S.; Madhavan, K.P., Modeling of ph process using wavenet based Hammerstein, J. Process Control, 17, 551-561, (2007)
[8] Veen, G.J.; Wingerden, J.W.; Fleming, P.A.; Scholbrock, A.K.; Verhaegen, M., Global data-driven modeling of wind turbines in the presence of turbulence, Control Eng. Pract., 21, 441-454, (2013)
[9] Moon, J.; Kim, B., Enhanced Hammerstein behavioral model for broadband wireless transmitters, IEEE Trans. Microw. Theory Tech., 59, 924-933, (2011)
[10] Le, F.; Markovsky, I.; Freeman, C.T.; Rogers, E., Recursive identification of Hammerstein systems with application to electrically stimulated muscle, Control Eng. Pract., 20, 386-396, (2012)
[11] Wang, J.; Sano, A.; Chen, T.; Huang, B., Identification of Hammerstein systems without explicit parameterization of nonlinearity, Int. J. Control, 82, 937-952, (2009) · Zbl 1165.93306
[12] Liu, Y.; Bai, E.W., Iterative identification of Hammerstein systems, Automatica, 43, 346-354, (2007) · Zbl 1111.93013
[13] Li, J.H., Parameter estimation for Hammerstein CARARMA systems based on the Newton iteration, Appl. Math. Lett., 26, 91-96, (2013) · Zbl 1255.65119
[14] Ding, F.; Chen, T., Identification of Hammerstein nonlinear ARMAX systems, Automatica, 41, 1479-1489, (2005) · Zbl 1086.93063
[15] Ding, F.; Shi, Y.; Chen, T., Gradient-based identification methods for Hammerstein nonlinear ARMAX models, Nonlinear Dyn., 45, 31-43, (2006) · Zbl 1134.93321
[16] Ding, F.; Shi, Y.; Chen, T., Auxiliary model based least-squares identification methods for Hammerstein output-error systems, Syst. Control Lett., 56, 373-380, (2007) · Zbl 1130.93055
[17] Vanbeylen, L.; Pintelon, R.; Schoukens, J., Blind maximum likelihood identification of Hammerstein systems, Automatica, 44, 3139-3146, (2008) · Zbl 1153.93522
[18] Wang, J.; Sano, A.; Shook, D.; Chen, T.; Huang, B., A blind approach to closed-loop identification of Hammerstein systems, Int. J. Control, 80, 302-313, (2007) · Zbl 1140.93350
[19] Vörös, J., Recursive identification of Hammerstein systems with discontinuous nonlinearities containing dead-zones, IEEE Trans. Autom. Control, 48, 2203-2206, (2003) · Zbl 1364.93172
[20] Vörös, J., Identification of nonlinear cascade systems with time-varying backlash, J. Electr. Eng., 62, 87-92, (2011)
[21] Wang, D.Q.; Chu, Y.Y.; Ding, F., Auxiliary model-based RELS and MI-ELS algorithms for Hammerstein OEMA systems, Comput. Math. Appl., 59, 3092-3098, (2010) · Zbl 1193.93170
[22] Wang, D.Q.; Chu, Y.Y.; Yang, G.W.; Ding, F., Auxiliary model-based recursive generalized least squares parameter estimation for Hammerstein OEAR systems, Math. Comput. Model., 52, 309-317, (2010) · Zbl 1201.93134
[23] Chen, J.; Lv, L.X.; Ding, R.F., Multi-innovation stochastic gradient algorithms for dual-rate sampled systems with preload nonlinearity, Appl. Math. Lett., 26, 124-129, (2013) · Zbl 1251.93130
[24] Li, J.H.; Ding, F., Maximum likelihood stochastic gradient estimation for Hammerstein systems with colored noise based on the key term separation technique, Comput. Math. Appl., 62, 4170-4177, (2011) · Zbl 1236.93150
[25] Li, J.H.; Ding, F.; Yang, G.W., Maximum likelihood least squares identification method for input nonlinear finite impulse response moving average systems, Math. Comput. Model., 55, 442-450, (2012) · Zbl 1255.93147
[26] Wang, W.; Ding, F.; Dai, J.Y., Maximum likelihood least squares identification for systems with autoregressive moving average noise, Appl. Math. Model., 36, 1842-1853, (2012) · Zbl 1242.62105
[27] Ding, F.; Liu, X.P.; Liu, G., Identification methods for Hammerstein nonlinear systems, Digit. Signal Process., 21, 215-238, (2011)
[28] Bai, E.W., An optimal two-stage identification algorithm for Hammerstein-Wiener nonlinear systems, Automatica, 34, 333-338, (1998) · Zbl 0915.93018
[29] Wang, D.Q.; Ding, F., Least squares based and gradient based iterative identification for Wiener nonlinear systems, Signal Process., 91, 1182-1189, (2011) · Zbl 1219.94052
[30] Ding, F.; Gu, Y., Performance analysis of the auxiliary model based least squares identification algorithm for one-step state delay systems, Int. J. Comput. Math., 89, 2019-2028, (2012) · Zbl 1255.93132
[31] Ding, F.; Gu, Y., Performance analysis of the auxiliary model-based stochastic gradient parameter estimation algorithm for state space systems with one-step state delay, Circuits Syst. Signal Process., 32, 585-599, (2013)
[32] Zhang, Y., Unbiased identification of a class of multi-input single-output systems with correlated disturbances using bias compensation methods, Math. Comput. Model., 53, 1810-1819, (2011) · Zbl 1219.93141
[33] Wang, D.Q.; Ding, F., Hierarchical least squares estimation algorithm for Hammerstein-Wiener systems, IEEE Signal Process. Lett., 19, 825-828, (2012)
[34] Ding, F., Hierarchical multi-innovation stochastic gradient algorithm for Hammerstein nonlinear system modeling, Appl. Math. Model., 37, 1694-1704, (2013) · Zbl 1349.93391
[35] Ding, F.; Chen, T., Hierarchical least squares identification methods for multivariable systems, IEEE Trans. Autom. Control, 50, 397-402, (2005) · Zbl 1365.93551
[36] Ding, F.; Chen, T., Hierarchical gradient-based identification of multivariable discrete-time systems, Automatica, 41, 315-325, (2005) · Zbl 1073.93012
[37] Dehghan, M.; Hajarian, M., An iterative method for solving the generalized coupled Sylvester matrix equations over generalized bisymmetric matrices, Appl. Math. Model., 34, 639-654, (2010) · Zbl 1185.65054
[38] Dehghan, M.; Hajarian, M., Analysis of an iterative algorithm to solve the generalized coupled Sylvester matrix equations, Appl. Math. Model., 35, 3285-3300, (2011) · Zbl 1227.65037
[39] Dehghan, M.; Hajarian, M., The reflexive and anti-reflexive solutions of a linear matrix equation and systems of matrix equations, Rocky Mt. J. Math., 40, 825-848, (2010) · Zbl 1198.15011
[40] Dehghan, M.; Hajarian, M., On the generalized bisymmetric and skew-symmetric solutions of the system of generalized Sylvester matrix equations, Linear Multilinear Algebra, 59, 1281-1309, (2011) · Zbl 1242.65075
[41] Dehghan, M.; Hajarian, M., The generalized centro-symmetric and least squares generalized centro-symmetric solutions of the matrix equation AYB+CYTD=\(E\), Math. Methods Appl. Sci., 34, (2011) · Zbl 1228.65066
[42] Dehghan, M.; Hajarian, M., Solving the generalized Sylvester matrix equation \(∑_{i=1}^{p} A_{i}XB_{i}+∑_{j=1}^{q}C_{j}YD_{j} = E\) over reflexive and anti-reflexive matrices, Int. J. Control. Autom. Syst., 9, 118-124, (2011)
[43] Dehghan, M.; Hajarian, M., Two algorithms for finding the Hermitian reflexive and skew-Hermitian solutions of Sylvester matrix equations, Appl. Math. Lett., 24, 444-449, (2011) · Zbl 1206.65144
[44] Dehghan, M.; Hajarian, M., The (\(R\),\(S\))-symmetric and (\(R\),\(S\))-skew symmetric solutions of the pair of matrix equations \(A\)_{1}XB_{1}=\(C\)_{1} and \(A\)_{2}XB_{2}=\(C\)_{2}, Bull. Iran. Math. Soc., 37, 269-279, (2011) · Zbl 1260.15022
[45] Dehghan, M.; Hajarian, M., SSHI methods for solving general linear matrix equations, Eng. Comput., 28, 1028-1043, (2012)
[46] Dehghan, M.; Hajarian, M., The generalised Sylvester matrix equations over the generalised bisymmetric and skew-symmetric matrices, Int. J. Syst. Sci., 43, 1580-1590, (2012) · Zbl 1308.65061
[47] Hashemi, B.; Dehghan, M., Efficient computation of enclosures for the exact solvents of a quadratic matrix equation, Electron. J. Linear Algebra, 20, 519-536, (2010) · Zbl 1207.65043
[48] Wang, D.Q., Least squares-based recursive and iterative estimation for output error moving average systems using data filtering, IET Control Theory Appl., 5, 1648-1657, (2011)
[49] Ding, F.; Liu, G.; Liu, X.P., Parameter estimation with scarce measurements, Automatica, 47, 1646-1655, (2011) · Zbl 1232.62043
[50] Ding, F.; Chen, T., Hierarchical identification of lifted state-space models for general dual-rate systems, IEEE Trans. Circuits Syst. I, Regul. Pap., 52, 1179-1187, (2005) · Zbl 1374.93342
[51] Goodwin, G.C., Sin, K.S.: Adaptive Filtering Prediction and Control. Prentice Hall, Englewood Cliffs (1984) · Zbl 0653.93001
[52] Shi, Y.; Yu, B., Output feedback stabilization of networked control systems with random delays modeled by Markov chains, IEEE Trans. Autom. Control, 54, 1668-1674, (2009) · Zbl 1367.93538
[53] Shi, Y.; Yu, B., Robust mixed H-2/H-infinity control of networked control systems with random time delays in both forward and backward communication links, Automatica, 47, 754-760, (2011) · Zbl 1215.93045
[54] Ding, F.; Qiu, L.; Chen, T., Reconstruction of continuous-time systems from their non-uniformly sampled discrete-time systems, Automatica, 45, 324-332, (2009) · Zbl 1158.93365
[55] Liu, Y.J.; Sheng, J.; Ding, R.F., Convergence of stochastic gradient estimation algorithm for multivariable ARX-like systems, Comput. Math. Appl., 59, 2615-2627, (2010) · Zbl 1193.60057
[56] Ding, F.; Liu, Y.J.; Bao, B., Gradient based and least squares based iterative estimation algorithms for multi-input multi-output systems, Proc. Inst. Mech. Eng., Part I, J. Syst. Control Eng., 226, 43-55, (2012)
[57] Ding, F., Coupled-least-squares identification for multivariable systems, IET Control Theory Appl., 7, 68-79, (2013)
[58] Ding, F., Two-stage least squares based iterative estimation algorithm for CARARMA system modeling, Appl. Math. Model., 37, 4798-4808, (2013) · Zbl 1438.93228
[59] Ding, F., Decomposition based fast least squares algorithm for output error systems, Signal Process., 93, 1235-1242, (2013)
[60] Ding, F.; Chen, T., Performance analysis of multi-innovation gradient type identification methods, Automatica, 43, 1-14, (2007) · Zbl 1140.93488
[61] Liu, Y.J.; Xiao, Y.S.; Zhao, X.L., Multi-innovation stochastic gradient algorithm for multiple-input single-output systems using the auxiliary model, Appl. Math. Comput., 215, 1477-1483, (2009) · Zbl 1177.65095
[62] Ding, F., Several multi-innovation identification methods, Digit. Signal Process., 20, 1027-1039, (2010)
[63] Ding, F.; Liu, X.P.; Liu, G., Multi-innovation least squares identification for linear and pseudo-linear regression models, IEEE Trans. Syst. Man Cybern., Part B, Cybern., 40, 767-778, (2010)
[64] Hu, P.P.; Ding, F., Multistage least squares based iterative estimation for feedback nonlinear systems with moving average noises using the hierarchical identification principle, Nonlinear Dyn., 73, 583-592, (2013) · Zbl 1281.93033
[65] Han, H.Q.; Xie, L.; Ding, F.; Liu, X., Hierarchical least squares based iterative identification for multivariable systems with moving average noises, Math. Comput. Model., 51, 1213-1220, (2010) · Zbl 1198.93216
[66] Zhang, Z.N.; Ding, F.; Liu, X.G., Hierarchical gradient based iterative parameter estimation algorithm for multivariable output error moving average systems, Comput. Math. Appl., 61, 672-682, (2011) · Zbl 1217.15022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.