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Liu, Wenxuan; Li, Meihang

Unbiased recursive least squares identification methods for a class of nonlinear systems with irregularly missing data. (English) Zbl 07844292

Int. J. Adapt. Control Signal Process. 37, No. 8, 2247-2275 (2023).
Summary: Missing data often occur in industrial processes. In order to solve this problem, an auxiliary model and a particle filter are adopted to estimate the missing outputs, and two unbiased parameter estimation methods are developed for a class of nonlinear systems (e.g., bilinear systems) with irregularly missing data. Firstly, an auxiliary model is constructed to estimate the unknown output, and an auxiliary model-based multi-innovation recursive least squares algorithm is presented by expanding the scalar innovation to an innovation vector. Secondly, according to the bias compensation principle, an auxiliary model-based bias compensation multi-innovation recursive least squares algorithm is proposed to compensate the bias caused by the colored noise. Thirdly, for further improving the parameter estimation accuracy, the unknown true output is estimated by a particle filter, and a particle filtering-based bias compensation multi-innovation recursive least squares algorithm is developed. Finally, a numerical example is selected to validate the effectiveness of the proposed algorithms. The simulation results indicate that the proposed algorithms have good performance in identifying bilinear systems with irregularly missing data.

Cited in 12 Documents

MSC:

93E24 Least squares and related methods for stochastic control systems
93E10 Estimation and detection in stochastic control theory
93E12 Identification in stochastic control theory

Keywords:

auxiliary model; bias compensation; bilinear system; parameter estimation; particle filtering
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References:

[1] YuWK, WuM, HuangB, et al. A generalized probabilistic monitoring model with both random and sequential data. Automatica. 2022;144:110468. · Zbl 1498.93708
[2] XuL. Separable multi‐innovation Newton iterative modeling algorithm for multi‐frequency signals based on the sliding measurement window. Circuits Syst Signal Process. 2022;41(2):805‐830. · Zbl 1509.94036
[3] XuL. Separable Newton recursive estimation method through system responses based on dynamically discrete measurements with increasing data length. Int J Control Autom Syst. 2022;20(2):432‐443.
[4] TerziE, FarinaM, FagianoL, et al. Robust multi‐rate predictive control using multi‐step prediction models learned from data. Automatica. 2022;136:109852. · Zbl 1480.93123
[5] ZhangB, SongY. Model‐predictive control for markovian jump systems under asynchronous scenario: an optimizing prediction dynamics approach. IEEE Trans Autom Control. 2022;67(9):4900‐4907. · Zbl 1537.93217
[6] YouJY, YuCP, SunJ, et al. Generalized maximum entropy based identification of graphical ARMA models. Automatica. 2022;141:110319. · Zbl 1491.93030
[7] DingF, LvL, PanJ, et al. Two‐stage gradient‐based iterative estimation methods for controlled autoregressive systems using the measurement data. Int J Control Autom Syst. 2020;18(4):886‐896.
[8] BalenzuelaMP, WillsAG, RentonC, et al. Parameter estimation for jump Markov linear systems. Automatica. 2022;135:109949. · Zbl 1478.93678
[9] WangJW, JiY, ZhangX, XuL. Two‐stage gradient‐based iterative algorithms for the fractional‐order nonlinear systems by using the hierarchical identification principle. Int J Adapt Control Signal Process. 2022;36(7):1778‐1796. · Zbl 07841875
[10] WangYJ, YangL. An efficient recursive identification algorithm for multilinear systems based on tensor decomposition. Int J Robust Nonlinear Control. 2021;31(16):7920‐7936. · Zbl 1527.93044
[11] DingF, MaH, PanJ, YangEF. Hierarchical gradient‐ and least squares‐based iterative algorithms for input nonlinear output‐error systems using the key term separation. J Frankl Inst. 2021;358(9):5113‐5135. · Zbl 1465.93040
[12] LiuXM, FanYM. Maximum likelihood extended gradient‐based estimation algorithms for the input nonlinear controlled autoregressive moving average system with variable‐gain nonlinearity. Int J Robust Nonlinear Control. 2021;31(9):4017‐4036. · Zbl 1526.93264
[13] KangZ, JiY, LiuXM. Hierarchical recursive least squares algorithms for Hammerstein nonlinear autoregressive output‐error systems. Int J Adapt Control Signal Process. 2021;35(11):2276‐2295. · Zbl 07840777
[14] WangJW, JiY, ZhangC. Iterative parameter and order identification for fractional‐order nonlinear finite impulse response systems using the key term separation. Int J Adapt Control Signal Process. 2021;35(8):1562‐1577. · Zbl 07840333
[15] WangYJ, TangSH, GuXB. Parameter estimation for nonlinear Volterra systems by using the multi‐innovation identification theory and tensor decomposition. J Frankl Inst. 2022;359(2):1782‐1802. · Zbl 1481.93022
[16] WangYJ, TangSH, DengMQ. Modeling nonlinear systems using the tensor network B‐spline and the multi‐innovation identification theory. Int J Robust Nonlinear Control. 2022;32(13):7304‐7318. · Zbl 1528.93083
[17] SchoukensM, TielsK. Identification of block‐oriented nonlinear systems starting from linear approximations: a survey. Automatica. 2017;85:272‐292. · Zbl 1375.93038
[18] JozsaM, PetreczkyM, CamlibelMK. Causality and network graph in general bilinear state‐space representations. IEEE Trans Autom Control. 2020;65(8):3623‐3630. · Zbl 1533.93262
[19] LiuSY. Iterative state and parameter estimation algorithms for bilinear state‐space systems by using the block matrix inversion and the hierarchical principle. Nonlinear Dyn. 2021;106(3):2183‐2202.
[20] GibsonS, WillsA, NinnessB. Maximum‐likelihood parameter estimation of bilinear systems. IEEE Trans Autom Control. 2005;50(10):1581‐1596. · Zbl 1365.93482
[21] DingF. Coupled‐least‐squares identification for multivariable systems. IET Control Theory Appl. 2013;7(1):68‐79.
[22] LiMH, LiuXM. Iterative identification methods for a class of bilinear systems by using the particle filtering technique. Int J Adapt Control Signal Process. 2021;35(10):2056‐2074. · Zbl 07840498
[23] DingF, LiuG, LiuXP. Partially coupled stochastic gradient identification methods for non‐uniformly sampled systems. IEEE Trans Autom Control. 2010;55(8):1976‐1981. · Zbl 1368.93121
[24] WangLJ, JiY. Decomposition‐based multiinnovation gradient identification algorithms for a special bilinear system based on its input‐output representation. Int J Robust Nonlinear Control. 2020;30(9):3607‐3623. · Zbl 1466.93172
[25] HafeziZ, ArefiMM. Recursive generalized extended least squares and RML algorithms for identification of bilinear systems with ARMA noise. ISA Trans. 2019;88:50‐61.
[26] LiuSY, ZhangX. Expectation‐maximization algorithm for bilinear systems by using the Rauch‐Tung‐Striebel smoother. Automatica. 2022;142:110365. · Zbl 1494.93030
[27] LiMH, LiuXM. The filtering‐based maximum likelihood iterative estimation algorithms for a special class of nonlinear systems with autoregressive moving average noise using the hierarchical identification principle. Int J Adapt Control Signal Process. 2019;33(7):1189‐1211. · Zbl 1425.93284
[28] XuL, ChenFY. Hierarchical recursive signal modeling for multi‐frequency signals based on discrete measured data. Int J Adapt Control Signal Process. 2021;35(5):676‐693. · Zbl 07839333
[29] XuL. Separable synchronous multi‐innovation gradient‐based iterative signal modeling from on‐line measurements. IEEE Trans Instrum Meas. 2022;71:6501313.
[30] YangXQ, YangXB. Local identification of LPV dual‐rate system with random measurement delays. IEEE Trans Ind Electron. 2018;65(2):1499‐1507.
[31] LiMH, LiuXM. Maximum likelihood hierarchical least squares‐based iterative identification for dual‐rate stochastic systems. Int J Adapt Control Signal Process. 2021;35(2):240‐261. · Zbl 07839197
[32] DingF, ChenT. Combined parameter and output estimation of dual‐rate systems using an auxiliary model. Automatica. 2004;40(10):1739‐1748. · Zbl 1162.93376
[33] DingF, ChenT. Parameter estimation of dual‐rate stochastic systems by using an output error method. IEEE Trans Autom Control. 2005;50(9):1436‐1441. · Zbl 1365.93480
[34] DingF, LiuXP, YangHZ. Parameter identification and intersample output estimation for dual‐rate systems. IEEE Trans Syst Man Cybern A Syst Humans. 2008;38(4):966‐975.
[35] DingF, DingJ. Least squares parameter estimation for systems with irregularly missing data. Int J Adapt Control Signal Process. 2010;24(7):540‐553. · Zbl 1200.93130
[36] LiJH, ZhengWX, GuJP, et al. Parameter estimation algorithms for Hammerstein output error systems using Levenberg‐Marquardt optimization method with varying interval measurements. J Frankl Inst. 2017;354(1):316‐331. · Zbl 1355.93180
[37] FanYM, LiuXM. Auxiliary model‐based multi‐innovation recursive identification algorithms for an input nonlinear controlled autoregressive moving average system with variable‐gain nonlinearity. Int J Adapt Control Signal Process. 2022;36(3):690‐707.
[38] XuL. Auxiliary model multiinnovation stochastic gradient parameter estimation methods for nonlinear sandwich systems. Int J Robust Nonlinear Control. 2021;31(1):148‐165. · Zbl 1525.93043
[39] LiMH, LiuXM. Maximum likelihood least squares based iterative estimation for a class of bilinear systems using the data filtering technique. Int J Control Autom Syst. 2020;18(6):1581‐1592.
[40] LiMH, LiuXM. Particle filtering‐based iterative identification methods for a class of nonlinear systems with interval‐varying measurements. Int J Control Autom Syst. 2022;20(7):2239‐2248.
[41] WeiC, ZhangX, XuL, et al. Overall recursive least squares and overall stochastic gradient algorithms and their convergence for feedback nonlinear controlled autoregressive systems. Int J Robust Nonlinear Control. 2022;32(9):5534‐5554. · Zbl 1528.93227
[42] GuY, ZhuQM, NouriH. Identification and U‐control of a state‐space system with time‐delay. Int J Adapt Control Signal Process. 2022;36(1):138‐154. · Zbl 07841277
[43] DingJL, ZhangWH. Finite‐time adaptive control for nonlinear systems with uncertain parameters based on the command filters. Int J Adapt Control Signal Process. 2021;35(9):1754‐1767. · Zbl 07840344
[44] WanLJ. Decomposition‐ and gradient‐based iterative identification algorithms for multivariable systems using the multi‐innovation theory. Circuits Syst Signal Process. 2019;38(7):2971‐2991.
[45] MaP, WangL. Filtering‐based recursive least squares estimation approaches for multivariate equation‐error systems by using the multiinnovation theory. Int J Adapt Control Signal Process. 2021;35(9):1898‐1915. · Zbl 1536.93999
[46] DingF. Least squares parameter estimation and multi‐innovation least squares methods for linear fitting problems from noisy data. J Comput Appl Math. 2023;426:115107. · Zbl 1518.93160
[47] LiuSY. Hierarchical principle‐based iterative parameter estimation algorithm for dual‐frequency signals. Circuits Syst Signal Process. 2019;38(7):3251‐3268.
[48] ChenJ, ZhuQM, LiuYJ. Modified Kalman filtering based multi‐step‐length gradient iterative algorithm for ARX models with random missing outputs. Automatica. 2020;118:109034. · Zbl 1447.93350
[49] HouJ, ChenFW, LiPH, ZhuZQ. Gray‐box parsimonious subspace identification of Hammerstein‐type systems. IEEE Trans Ind Electron. 2021;68(10):9941‐9951.
[50] ZhangX. State estimation for bilinear systems through minimizing the covariance matrix of the state estimation errors. Int J Adapt Control Signal Process. 2019;33(7):1157‐1173. · Zbl 1425.93278
[51] JiY, JiangAN. Filtering‐based accelerated estimation approach for generalized time‐varying systems with disturbances and colored noises. IEEE Trans Circuits Syst II Express Briefs. 2023;70(1):206‐210.
[52] XiongJX, PanJ, ChenGY, et al. Sliding mode dual‐channel disturbance rejection attitude control for a quadrotor. IEEE Trans Ind Electron. 2022;69(10):10489‐10499.
[53] PanJ, LiW, ZhangHP. Control algorithms of magnetic suspension systems based on the improved double exponential reaching law of sliding mode control. Int J Control Autom Syst. 2018;16(6):2878‐2887.
[54] XuL. Decomposition strategy‐based hierarchical least mean square algorithm for control systems from the impulse responses. Int J Syst Sci. 2021;52(9):1806‐1821. · Zbl 1483.93637
[55] PanJ, ChenQ, XiongJ, ChenG. A novel quadruple boost nine level switched capacitor inverter. J Electr Eng Technol. 2023;18(1):467‐480.
[56] XuC, QinY, SuH. Observer‐based dynamic event‐triggered bipartite consensus of discrete‐time multi‐agent systems. IEEE Trans Circuits Syst II Express Briefs. 2023;70(3):1054‐1058.
[57] ZhouYH. Partially‐coupled nonlinear parameter optimization algorithm for a class of multivariate hybrid models. Appl Math Comput. 2022;414:126663. · Zbl 1510.93338
[58] GengFZ, WuXY. A novel kernel functions algorithm for solving impulsive boundary value problems. Appl Math Lett. 2022;134:108318. · Zbl 1503.65151
[59] LiXY, WuBY. A kernel regression approach for identification of first order differential equations based on functional data. Appl Math Lett. 2022;127:107832. · Zbl 1490.34021
[60] WangHJ, KeGY, PanJ, et al. Two pairs of heteroclinic orbits coined in a new sub‐quadratic Lorenz‐like system. Eur Phys J B. 2023;96(3):28.
[61] WangHJ, HeGY, DongGL, et al. Singularly degenerate heteroclinic cycles with nearby apple‐shape attractors. Int J Bifurc Chaos. 2023;33(1):2350011. · Zbl 07847280
[62] WangH, FanH, PanJ. A true three‐scroll chaotic attractor coined. Discr Contin Dyn Syst Ser B. 2022;27(5):2891‐2915. · Zbl 1495.34025
[63] DingF, LiuYJ, BaoB. Gradient based and least squares based iterative estimation algorithms for multi‐input multi‐output systems. Proc Inst Mech Eng I J Syst Control Eng. 2012;226(1):43‐55.
[64] PanJ, LiuYQ, ShuJ. Gradient‐based parameter estimation for an exponential nonlinear autoregressive time‐series model by using the multi‐innovation. Int J Control Autom Syst. 2023;21(1):140‐150.
[65] ZhouYH. Hierarchical estimation approach for RBF‐AR models with regression weights based on the increasing data length. IEEE Trans Circuits Syst II Express Briefs. 2021;68(12):3597‐3601.
[66] PanJ, JiangX, WanXK, DingW. A filtering based multi‐innovation extended stochastic gradient algorithm for multivariable control systems. Int J Control Autom Syst. 2017;15(3):1189‐1197.
[67] JiY, ZhangC, KangZ, YuT. Parameter estimation for block‐oriented nonlinear systems using the key term separation. Int J Robust Nonlinear Control. 2020;30(9):3727‐3752. · Zbl 1466.93161
[68] MaH, PanJ, DingW. Partially‐coupled least squares based iterative parameter estimation for multi‐variable output‐error‐like autoregressive moving average systems. IET Control Theory Appl. 2019;13(18):3040‐3051.
[69] LiJM. A novel nonlinear optimization method for fitting a noisy Gaussian activation function. Int J Adapt Control Signal Process. 2022;36(3):690‐707. · Zbl 07841310
[70] CaoY, AnY, SuS, et al. A statistical study of railway safety in China and Japan 1990‐2020. Accid Anal Prev. 2022;175:106764.
[71] CaoY, YangR, MaL, WenJ. Research on virtual coupled train control method based on GPC & VAPF. Chin J Electron. 2022;31(5):897‐905.
[72] SunY, CaoY, LiP. Contactless fault diagnosis for railway point machines based on multi‐scale fractional wavelet packet energy entropy and synchronous optimization strategy. IEEE Trans Veh Technol. 2022;71(6):5906‐5914.
[73] CuiT. Moving data window‐based partially‐coupled estimation approach for modeling a dynamical system involving unmeasurable states. ISA Trans. 2022;128:437‐452.
[74] CaoY, JiY, SunY, SuS. The fault diagnosis of a switch machine based on deep random forest fusion. IEEE Intell Transp Syst Mag. 2023;15(1):437‐452.
[75] WangX, SuS, CaoY, WangXL. Robust control for dynamic train regulation in fully automatic operation system under uncertain wireless transmissions. IEEE Trans Intell Transp Syst. 2022;23(11):20721‐20734.
[76] JiY, KangZ, ZhangX, et al. Model recovery for multi‐input signal‐output nonlinear systems based on the compressed sensing recovery theory. J Frankl Inst. 2022;359(5):2317‐2339. · Zbl 1485.93281
[77] MaH. A novel multi‐innovation gradient support vector machine regression method. ISA Trans. 2022;130:343‐359.
[78] JiY, KangZ. Three‐stage forgetting factor stochastic gradient parameter estimation methods for a class of nonlinear systems. Int J Robust Nonlinear Control. 2021;31(3):971‐987. · Zbl 1525.93438
[79] PanJ, LiuSD, ShuJ, WanXK. Hierarchical recursive least squares estimation algorithm for secondorder Volterra nonlinear systems. Int J Control Autom Syst. 2022;20(12):3940‐3950.
[80] JiY, KangZ, LiuXM. The data filtering based multiple‐stage Levenberg‐Marquardt algorithm for Hammerstein nonlinear systems. Int J Robust Nonlinear Control. 2021;31(15):7007‐7025. · Zbl 1527.93457
[81] PanJ, MaH, ZhangX, et al. Recursive coupled projection algorithms for multivariable output‐error‐like systems with coloured noises. IET Signal Process. 2020;14(7):455‐466.
[82] JiY, JiangXK, WanLJ. Hierarchical least squares parameter estimation algorithm for two‐input Hammerstein finite impulse response systems. J Frankl Inst. 2020;357(8):5019‐5032. · Zbl 1437.93131
[83] ZhangX. Hierarchical parameter and state estimation for bilinear systems. Int J Syst Sci. 2020;51(2):275‐290. · Zbl 1483.93677
[84] DingF, XuL, ZhangX, ZhouYH. Filtered auxiliary model recursive generalized extended parameter estimation methods for box‐Jenkins systems by means of the filtering identification idea. Int J Robust Nonlinear Control. 2023;33(10):5510‐5535. · Zbl 1532.93369
[85] ShiZW, YangHD, DaiM. The data‐filtering based bias compensation recursive least squares identification for multi‐input single‐output systems with colored noises. J Frankl Inst. 2023;360(7):4753‐4783. · Zbl 1516.93122
[86] PanJ, ZhangH, GuoH, et al. Multivariable CAR‐like system identification with multi‐innovation gradient and least squares algorithms. Int J Control Autom Syst. 2023;21:1455‐1464.
[87] Xu L. Parameter estimation for nonlinear functions related to system responses. Int J Control Autom Syst. 2023;21(6):1780‐1792.
[88] Xu L. Separable synthesis estimation methods and convergence analysis for multivariable systems. J Comput Appl Math. 2023;427:115104. · Zbl 1518.93138
[89] Pan J, Shao B, Xiang JX. Attitude control of quadrotor UAVs based on adaptive sliding mode. Int J Control Autom Syst. 2023;21. http://dx.doi.org/10.1007/s12555‐022‐0189‐2
[90] Zhang X. Highly computationally efficient state filter based on the delta operator. Int J Adapt Control Signal Process. 2019; 33(6):875‐889. · Zbl 1425.93290
[91] Xu H. Joint parameter and time‐delay estimation for a class of nonlinear time‐series models. IEEE Signal Process Lett. 2022;29:947‐951.
[92] Geng FZ, Wu XY. Reproducing kernel‐based piecewise methods for efficiently solving oscillatory systems of second‐order initial value problems. Calcolo. 2023;60(2):20. · Zbl 1517.65059
[93] Zhao SY, Li K. Tuning‐free Bayesian estimation algorithms for faulty sensor signals in state‐space. IEEE Trans Ind Electron. 2023;70(1):921‐929.
[94] Zhao SY, Shmaliy YS. Batch optimal FIR smoothing: increasing state informativity in nonwhite measurement noise environments. IEEE Trans Ind Inf. 2023. doi:10.1109/tii.2022.3193879.
[95] Zhao SY, Wang JF. Discrete time q‐lag maximum likelihood FIR smoothing and iterative recursive algorithm. IEEE Trans Signal Process. 2021;69:6342‐6354. · Zbl 07908798
[96] Zhang TY, Zhao SY. Bayesian inference for state‐space models with Student‐t mixture distributions. IEEE Trans Cybern. 2023. doi:10.1109/tcyb.2022.3183104.
[97] Zhang X. Optimal adaptive filtering algorithm by using the fractional‐order derivative. IEEE Signal Process Lett. 2022; 29:399‐403.
[98] Zhang X. Adaptive parameter estimation for a general dynamical system with unknown states. Int J Robust Nonlinear Control. 2020;30(4):1351‐1372. · Zbl 1465.93115
[99] Zhao SY, Huang B. Online probabilistic estimation of sensor faulty signal in industrial processes and its applications. IEEE Trans Ind Electron. 2021;68(9):8858‐8862.
[100] Zhao SY, Shmaliy YS. Multipass optimal FIR filtering for processes with unknown initial states and temporary mismatches. IEEE Trans Ind Inf. 2021;17(8):5360‐5368.
[101] Zhou YH. Modeling nonlinear processes using the radial basis function‐based state‐dependent autoregressive models. IEEE Signal Process Lett. 2020;27:1600‐1604.
[102] Zhang X. Recursive parameter estimation methods and convergence analysis for a special class of nonlinear systems. Int J Robust Nonlinear Control. 2020;30(4):1373‐1393. · Zbl 1465.93218
[103] Zhao SY, Shmaliy YS. Self‐tuning unbiased finite impulse response filtering algorithm for processes with unknown measurement noise covariance. IEEE Trans Control Syst Technol. 2021;29(3):1372‐1379.
[104] Zhao SY, Huang B. Trial‐and‐error or avoiding a guess? Initialization of the Kalman filter. Automatica. 2020;121:109184. · Zbl 1448.93335
[105] Zhao SY, Shmaliy YS. An improved iterative FIR state estimator and its applications. IEEE Trans Ind Inf. 2020;16(2):1003‐1012.
[106] Zhao SY, Shmaliy YS. Probabilistic monitoring of correlated sensors for nonlinear processes in state space. IEEE Trans Ind Electron. 2020;67(3):2294‐2303.
[107] Ding F, Yang HZ, Liu F. Performance analysis of stochastic gradient algorithms under weak conditions. Science in China Series F‐‐Information Sciences. 2008;51(9):1269‐1280. · Zbl 1145.93050
[108] Ding F, Liu XM, Chen HB, Yao GY. Hierarchical gradient based and hierarchical least squares based iterative parameter identification for CARARMA systems. Signal Process. 2014;97:31‐39.
[109] Cao Y, Zhang Z. Trajectory optimization for high‐speed trains via a mixed integer linear programming approach. IEEE Trans Intell Transp Syst. 2022;23(10), 17666‐17676.
[110] Cao Y, Wen JK. Parameter‐varying artificial potential field control of virtual coupling system with nonlinear dynamics. Fractals. 2022;30(2):2240099. · Zbl 1485.93023
[111] Ding F, Chen T, Iwai Z. Adaptive digital control of Hammerstein nonlinear systems with limited output sampling. SIAM J Control Optim. 2007;45(6):2257‐2276. · Zbl 1126.93034
[112] Ding F, Shi Y, Chen T. Performance analysis of estimation algorithms of non‐stationary ARMA processes. IEEE Trans Signal Process. 2006;54(3):1041‐1053. · Zbl 1373.94569
[113] Cao Y, Wen J. Tracking and collision avoidance of virtual coupling train control system. Alex Eng J. 2021;60(2):2115‐2125.
[114] Cao Y, Sun Y. A sound‐based fault diagnosis method for railway point machines based on two‐stage feature selection strategy and ensemble classifier. IEEE Trans Intell Transp Syst. 2022;23(8):12074‐12083.
[115] Ding F, Chen T, Qiu L. Bias compensation based recursive least squares identification algorithm for MISO systems. IEEE Trans Circuits Syst II Express Briefs. 2006;53(5):349‐353.
[116] Wang YJ. Novel data filtering based parameter identification for multiple‐input multiple‐output systems using the auxiliary model. Automatica. 2016;71:308‐313. · Zbl 1343.93087
[117] Liu YJ. An efficient hierarchical identification method for general dual‐rate sampled‐data systems. Automatica. 2014;50(3):962‐970. · Zbl 1298.93227
[118] Ding J. Hierarchical least squares identification for linear SISO systems with dual‐rate sampled‐data. IEEE Trans Autom Control. 2011;56(11):2677‐2683. · Zbl 1368.93744
[119] Wang YJ. Recursive parameter estimation algorithm for multivariate output‐error systems. J Frankl Inst. 2018; 355(12):5163‐5181. · Zbl 1395.93288
[120] Hu C, Ji Y, Ma CQ. Joint two‐stage multi‐innovation recursive least squares parameter and fractional‐order estimation algorithm for the fractional‐order input nonlinear output‐error autoregressive model. Int J Adapt Control Signal Process. 2023;37. doi:10.1002/acs.3593 · Zbl 07844262
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.