Restricted accessResearch articleFirst published online 2025-3
Auxiliary model maximum likelihood least squares-based iterative algorithm for multivariable autoregressive output-error autoregressive moving average systems
Identification of multivariable systems is of great significance to control systems. This paper focuses on the parameter identification problems for multivariable autoregressive output-error autoregressive moving average (M-AROEARMA) systems. On the basis of the decomposition strategy, the M-AROEARMA model is de- composed into multiple subsystem models. By means of the auxiliary model idea, the auxiliary model least squares-based iterative algorithm is derived. For the purpose of achieving highly accurate parameter identification performance under colored noises interference, an auxiliary model maximum likelihood least squares-based iterative algorithm is proposed by utilizing the maximum likelihood principle. The numerical simulation example demonstrates the effectiveness of the proposed algorithms.
LiuSYinXLiuJ.Sensor network design for post-combustion CO2 capture plants: computational efficiency and robustness. J Process Control2023; 129: 103035.
2.
ThomasMJGeorgeSSreedharanD, et al. Dynamic modeling, system identification and comparative study of various control strategies for a spatial parallel manipulator. Proc IMechE, Part I: J Systems and Control Engineering2022; 236(2): 270–293.
3.
DingF.Least squares parameter estimation and multi-innovation least squares methods for linear fitting problems from noisy data. J Comput Appl Math2023; 426: 115107.
4.
MarzouguiSBedouiSAbderrahimK.On the combined estimation of the parameters and the states of fractional-order systems. Proc IMechE, Part I: J Systems and Control Engineering2023; 237(10): 1853–1866.
5.
XuL.Separable Newton recursive estimation method through system responses based on dynamically discrete measurements with increasing data length. Int J Control Autom Syst2022; 20(2): 432–443.
6.
AnSHeYWangL.Maximum likelihood based multi-innovation stochastic gradient identification algorithms for bilinear stochastic systems with ARMA noise. Int J Adapt Control Signal Process2023; 37(10): 2690–2705.
7.
WangXMaJXiongW.Expectation-maximization algorithm for bilinear state-space models with time-varying delays under non-Gaussian noise. Int J Adapt Control Signal Process2023; 37(10): 2706–2724.
8.
JiYKangZ.Three-stage forgetting factor stochastic gradient parameter estimation methods for a class of nonlinear systems. Int J Robust Nonlinear Control2021; 31(3): 971–987.
9.
WangYDingFWuM.Recursive parameter estimation algorithm for multivariate output-error systems. J Franklin Inst2018; 355(12): 5163–5181.
10.
CoxPBTóthR.Linear parameter-varying subspace identification: a unified framework. Automatica2021; 123: 109296.
11.
KaracanSHapoğluHAlpbazM.Multivariable system identification and generic model control of a laboratory scale packed distillation column. Appl Therm Eng2007; 27(5-6): 1017–1028.
12.
ZhouYDingF.A novel recursive multivariate nonlinear time-series modeling method by using the coupling identification concept. Appl Math Model2024; 127: 571–587.
13.
AzzouzMAEl-SaadanyEF.Multivariable grid admittance identification for impedance stabilization of active distribution networks. IEEE Trans Smart Grid2017; 8(3): 1116–1128.
14.
XingHDingFPanF, et al. Hierarchical recursive least squares parameter estimation methods for multiple-input multiple-output systems by using the auxiliary models. Int J Adapt Control Signal Process2023; 37(11): 2983–3007.
15.
XiaHJiYXuL, et al. Maximum likelihood-based recursive least-squares algorithm for multivariable systems with colored noises using the decomposition technique. Circuits Syst Signal Process2019; 38(3): 986–1004.
16.
XingHDingFPanF.Highly-computational hierarchical iterative identification methods for multiple-input multiple-output systems by using the auxiliary models. Int J Robust Nonlinear Control2023; 33(17): 10845–10863.
17.
XingHDingFZhangX, et al. Highly-efficient filtered hierarchical identification algorithms for multiple-input multiple-output systems with colored noises. Syst Control Lett2024; 186; 105762.
DingFChenT.Combined parameter and output estimation of dual-rate systems using an auxiliary model. Automatica2004; 40(10): 1739–1748.
20.
DingFChenT.Parameter estimation of dual-rate stochastic systems by using an output error method. IEEE Trans Automat Contr2005; 50(9): 1436–1441.
21.
MaoYXuCChenJ, et al. Auxiliary model-based iterative estimation algorithms for nonlinear systems using the covariance matrix adaptation strategy. Circuits Syst Signal Process2022; 41(12): 6750–6773.
22.
AnSWangLHeY, et al. The auxiliary model based hierarchical estimation algorithms for wave peak frequency identification. Int J Control Autom Syst2023; 21(3): 745–754.
23.
FanYLiuX.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 Process2022; 36(3): 521–540.
24.
LiuJJiY.Auxiliary model-based recursive least squares algorithm for two-input single-output Hammerstein output-error moving average systems by using the hierarchical identification principle. Int J Robust Nonlinear Control2022; 32(13): 7575–7593.
25.
RamadanMSBitmeadRR.Maximum likelihood recursive state estimation using the expectation maximization algorithm. Automatica2022; 144: 110482.
26.
WangLAnSHeY, et al. The filtering based maximum likelihood recursive least squares parameter estimation algorithms for a class of nonlinear stochastic systems with colored noise. Int J Control Autom Syst2023; 21(1): 151–160.
27.
LiSJiY.Maximum likelihood interval-varying recursive least squares identification for output-error autoregressive systems with scarce measurements. J Franklin Inst2023; 360(11): 7230–7246.
28.
ZongTLiJLuG.Hierarchical maximum likelihood identification based on Levenberg–Marquardt and particle swarm optimization for feedback nonlinear systems. Proc IMechE, Part I: J Systems and Control Engineering2023; 237(9): 1573–1586.
29.
LiMLiuX.Maximum likelihood least squares based iterative estimation for a class of bilinear systems using the data filtering technique. Int J Control Autom Syst2020; 18(6): 1581–1592.
30.
LiuHWangJJiY.Maximum likelihood recursive generalized extended least squares estimation methods for a bilinear-parameter systems with ARMA noise based on the over-parameterization model. Int J Control Autom Syst2022; 20(8): 2606–2615.
31.
OrellanaRBittnerGCarvajalR, et al. Maximum likelihood estimation for non-minimum-phase noise transfer function with Gaussian mixture noise distribution. Automatica2022; 135: 109937.
32.
HuCJiYMaC.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 Process2023; 37(7): 1650–1670.
33.
TimoudasTOZhangSMagnússonS, et al. A general framework to distribute iterative algorithms with localized information over networks. IEEE Trans Automat Contr2023; 68(12): 7358–7373.
34.
ChenJZhuQLiuY.Modified Kalman filtering based multi-step-length gradient iterative algorithm for ARX models with random missing outputs. Automatica2020; 118: 109034.
35.
DingFXuLZhangX, et al. Hierarchical gradient- and least-squares-based iterative estimation algorithms for input-nonlinear output-error systems from measurement information by using the over-parameterization. Int J Robust Nonlinear Control2024; 34(2): 1120–1147.
36.
WuXYangXHuangJ, et al. Remaining useful life prediction for motor systems by iteratively updated wiener process based on closed-loop performance degradation monitoring. Proc IMechE, Part I: J Systems and Control Engineering2024; 238(1): 3–15.
37.
PasqualettiFCarliRBulloF.Distributed estimation via iterative projections with application to power network monitoring. Automatica2012; 48(5): 747–758.
38.
WangDFanQMaY.An interactive maximum likelihood estimation method for multivariable Hammerstein systems. J Franklin Inst2020; 357(17): 12986–13005.
39.
YouJLiuY.Iterative identification for multivariable systems with time-delays based on basis pursuit de-noising and auxiliary model. Algorithms2018; 11(11): 180.
40.
JafariMSalimifardMDehghaniM.Identification of multivariable nonlinear systems in the presence of colored noises using iterative hierarchical least squares algorithm. ISA Trans2014; 53(4): 1243–1252.
41.
WangJJiYZhangX, et al. Two-stage gradient-based iterative algorithms for the fractional-order nonlinear systems by using the hierarchical identification principle. Int J Adapt Control Signal Process2022; 36(7): 1778–1796.
42.
DingF.System identification-performances analysis for identification methods. Beijing: Science Press, 2014.
43.
DingF.System identification-multi-innovation identification theory and methods. Beijing: Science Press, 2016.
44.
DingF.System identification-auxiliary model identification idea and methods. Beijing: Science Press, 2017.
PanJZhangHGuoH, et al. Multivariable CAR-like system identification with multi-innovation gradient and least squares algorithms. Int J Control Autom Syst2023; 21(5): 1455–1464.
47.
WangYDingF.Filtering-based iterative identification for multivariable systems. IET Control Theory Appl2016; 10(8): 894–902.
48.
JiYJiangA.Filtering-based accelerated estimation approach for generalized time-varying systems with disturbances and colored noises. IEEE Trans Circuits Syst II Express Briefs2023; 70(1): 206–210.
49.
LeeKWSinghSN.Composite adaptive control of submarine and parameter identification by integral regressor excitation. Proc IMechE, Part I: J Systems and Control Engineering2024; 238(5): 907–928.
50.
DingFXuLZhangX, et al. Filtered auxiliary model recursive generalized extended parameter estimation methods for Box–Jenkins systems by means of the filtering identification idea. Int J Robust Nonlinear Control2023; 33(10): 5510–5535.
51.
GuptaSPadheeSPekarL.Recursive least squares identification of heat exchanger system using block-structured models. Proc IMechE, Part I: J Systems and Control Engineering2022; 236(4): 870–879.
52.
LiuSZhangXXuL, et al. Expectation–maximization algorithm for bilinear systems by using the Rauch–Tung–Striebel smoother. Automatica2022; 142: 110365.
53.
WuX.Data filtering based multi-innovation gradient identification for bilinear systems with colored noise. Int J Adapt Control Signal Process2023; 37(7): 1940–1960.
54.
GuYDaiWZhuQ, et al. Hierarchical multi-innovation stochastic gradient identification algorithm for estimating a bilinear state-space model with moving average noise. J Comput Appl Math2023; 420: 114794.
55.
LiuSWangYDingF, et al. Joint iterative state and parameter estimation for bilinear systems with autoregressive noises via the data filtering. ISA Trans2024; 147: 337–349.
56.
DingFXuLZhangX, et al. Recursive identification methods for general stochastic systems with colored noises by using the hierarchical identification principle and the filtering identification idea. Annu Rev Control2024; 57: 100942.
57.
JiYZhangCKangZ, et al. Parameter estimation for block-oriented nonlinear systems using the key term separation. Int J Robust Nonlinear Control2020; 30(9): 3727–3752.
58.
HouJChenFLiP, et al. Gray-box parsimonious subspace identification of Hammerstein-type systems. IEEE Trans Ind Electron2021; 68(10): 9941–9951.
59.
YangDDingF.Multi-innovation gradient-based iterative identification methods for feedback nonlinear systems by using the decomposition technique. Int J Robust Nonlinear Control2023; 33(13): 7755–7773.
60.
YangDLiuYDingF, et al. Hierarchical gradient-based iterative parameter estimation algorithms for a nonlinear feedback system based on the hierarchical identification principle. Circuits Syst Signal Process2024; 43(1): 124–151.
61.
XuL.Parameter estimation for nonlinear functions related to system responses. Int J Control Autom Syst2023; 21(6): 1780–1792.
62.
WanLDingF.Decomposition- and gradient-based iterative identification algorithms for multivariable systems using the multi-innovation theory. Circuits Syst Signal Process2019; 38(7): 2971–2991.
63.
DingFShaoXXuL, et al. Filtered generalized iterative parameter identification for equation-error autoregressive models based on the filtering identification idea. Int J Adapt Control Signal Process2024; 38(4): 1363–1385.
64.
ZhangXDingF.Adaptive parameter estimation for a general dynamical system with unknown states. Int J Robust Nonlinear Control2020; 30(4): 1351–1372.
65.
ZhangXDingF.Hierarchical parameter and state estimation for bilinear systems. Int J Syst Sci2020; 51(2): 275–290.
66.
ZhangXDingFXuL, et al. Highly computationally efficient state filter based on the delta operator. Int J Adapt Control Signal Process2019; 33(6): 875–889.
67.
LiMLiuXDingF.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 Process2019; 33(7): 1189–1211.
68.
ZhangXDingFYangE.State estimation for bilinear systems through minimizing the covariance matrix of the state estimation errors. Int J Adapt Control Signal Process2019; 33(7): 1157–1173.
69.
ZhangXDingFXuL.Recursive parameter estimation methods and convergence analysis for a special class of nonlinear systems. Int J Robust Nonlinear Control2020; 30(4): 1373–1393.
70.
LiuHWangJMengX.Hierarchical maximum likelihood generalized extended stochastic gradient algorithms for bilinear-in-parameter systems. Optim Control Appl Methods2022; 43(2): 402–417.
71.
JiYKangZZhangX, et al. Model recovery for multi-input signal-output nonlinear systems based on the compressed sensing recovery theory. J Franklin Inst2022; 359(5): 2317–2339.
72.
JiYKangZLiuX.The data filtering based multiple-stage Levenberg–Marquardt algorithm for Hammerstein nonlinear systems. Int J Robust Nonlinear Control2021; 31(15): 7007–7025.
73.
XuHDingFChampagneB.Joint parameter and time-delay estimation for a class of nonlinear time-series models. IEEE Signal Process Lett2022; 29: 947–951.
74.
ZhouYZhangXDingF.Hierarchical estimation approach for RBF-AR models with regression weights based on the increasing data length. IEEE Trans Circuits Syst II Express Briefs2021; 68(12): 3597–3601.
75.
DingFMaHPanJ, et al. Hierarchical gradient- and least squares-based iterative algorithms for input nonlinear output-error systems using the key term separation. J Franklin Inst2021; 358(9): 5113–5135.
76.
XuLDingF.Decomposition and composition modeling algorithms for control systems with colored noises. Int J Adapt Control Signal Process2024; 38(1): 255–278.
77.
BiYJiY.Parameter estimation of fractional-order Hammerstein state space system based on the extended Kalman filter. Int J Adapt Control Signal Process2023; 37(7): 1827–1846.
78.
ZhaoSShmaliyYSAhnCK, et al. Probabilistic monitoring of correlated sensors for nonlinear processes in state space. IEEE Trans Ind Electron2020; 67(3): 2294–2303.
79.
XuLDingF.Separable synthesis gradient estimation methods and convergence analysis for multivariable systems. J Comput Appl Math2023; 427: 115104.
80.
ChenJPuYGuoL, et al. Second-order optimization methods for time-delay autoregressive eXogenous models: nature gradient descent method and its two modified methods. Int J Adapt Control Signal Process2023; 37(1): 211–223.
81.
WeiCZhangXXuL, et al. Overall recursive least squares and overall stochastic gradient algorithms and their convergence for feedback nonlinear controlled autoregressive systems. Int J Robust Nonlinear Control2022; 32(9): 5534–5554.
82.
XuLXuHDingF.Adaptive multi-innovation gradient identification algorithms for a controlled autoregressive autoregressive moving average model. Circuits Syst Signal Process2024; 43(6): 3718–3747.
83.
WangYYangL.An efficient recursive identification algorithm for multilinear systems based on tensor decomposition. Int J Robust Nonlinear Control2021; 31(16): 7920–7936.
84.
LiJDingFHayatT.A novel nonlinear optimization method for fitting a noisy Gaussian activation function. Int J Adapt Control Signal Process2022; 36(3): 690–707.
85.
JiYLiuJLiuH.An identification algorithm of generalized time-varying systems based on the Taylor series expansion and applied to a pH process. J Process Control2023; 128: 103007.
86.
MiaoGDingFLiuQ, et al. Iterative parameter identification algorithms for transformed dynamic rational fraction input–output systems. J Comput Appl Math2023; 434: 115297.
87.
CaoYSunYXieG, et al. Fault diagnosis of train plug door based on a hybrid criterion for IMFs selection and fractional wavelet package energy entropy. IEEE Trans Vehicular Technol2019; 68(8): 7544–7551.
88.
CaoYSunYLiP, et al. Vibration-based fault diagnosis for railway point machines using multi-domain features, ensemble feature selection and SVM. IEEE Trans Vehicular Technol2024; 73(1): 176–184.
89.
CaoYLiuYSunY, et al. Enhancing rail safety through real-time defect detection: a novel lightweight network approach. Accid Anal Prev2024; 203: 107617.
90.
ZhouYLingKVDingF, et al. Online network-based identification and its application in satellite attitude control systems. IEEE Trans Aerosp Electron Syst2023; 59(3): 2530–2543.
91.
CaoYMaLXiaoS, et al. Standard analysis for transfer delay in CTCS-3. Chin J Electron2017; 26(5): 1057–1063.
92.
SunYCaoYLiP.Contactless fault diagnosis for railway point machines based on multi-scale fractional wavelet packet energy entropy and synchronous optimization strategy. IEEE Trans Vehicular Technol2022; 71(6): 5906–5914.
93.
XuLDingFLuX, et al. Hierarchical multi-innovation generalised extended stochastic gradient methods for multivariable equation-error autoregressive moving average systems. IET Control Theory Appl2020; 14(10): 1276–1286.
94.
HuCLiuHJiY.Parameter and order estimation algorithms and convergence analysis for lithium-ion batteries. Int J Robust Nonlinear Control2023; 33(18): 11411–11433.
95.
XuLDingFYangE.Separable recursive gradient algorithm for dynamical systems based on the impulse response signals. Int J Control Autom Syst2020; 18(12): 3167–3177.
96.
HouJ.Parsimonious model based consistent subspace identification of hammerstein systems under periodic disturbances. Int J Control Autom Syst2024; 22(1): 61–71.
97.
HouJWangHSuH, et al. A bias-correction modeling method of Hammerstein–Wiener systems with polynomial nonlinearities using noisy measurements. Mech Syst Signal Process2024; 213: 111329.
98.
LiuQChenF.Model transformation based distributed stochastic gradient algorithm for multivariate output-error systems. Int J Syst Sci2023; 54(7): 1484–1502.
99.
CaoYWenJHobinyA, et al. Parameter-varying artificial potential field control of virtual coupling system with nonlinear dynamics. Fractals2022; 30(02): 2240099.
100.
CaoYWenJMaL.Tracking and collision avoidance of virtual coupling train control system. Alex Eng J2021; 60(2): 2115–2125.
101.
SuSWangXCaoY, et al. An energy-efficient train operation approach by integrating the metro timetabling and eco-driving. IEEE Trans Intell Transp Syst2020; 21(10): 4252–4268.
102.
SuSZhuQLiuJ, et al. A data-driven iterative learning approach for optimizing the train control strategy. IEEE Trans Ind Inform2023; 19(7): 7885–7893.
103.
SuSSheJLiK, et al. A nonlinear safety equilibrium spacing-based model predictive control for virtually coupled train set over gradient terrains. IEEE Trans Transp Electrification2022; 8(2): 2810–2824.
104.
SuSTangTXunJ, et al. Design of running grades for energy-efficient train regulation: a case study for Beijing Yizhuang line. IEEE Intell Transp Syst Mag2021; 13(2): 189–200.
105.
CaoYAnYSuS, et al. A statistical study of railway safety in China and Japan 1990–2020. Accid Anal Prev2022; 175: 106764.
106.
CaoYYangYMaL, et al. Research on virtual coupled train control method based on GPC & VAPF. Chin J Electron2022; 31(5): 897–905.
107.
WangXSuSCaoY, et al. Robust control for dynamic train regulation in fully automatic operation system under uncertain wireless transmissions. IEEE Trans Intell Transp Syst2022; 23(11): 20721–20734.
108.
CaoYZhangZChengF, et al. Trajectory optimization for high-speed trains via a mixed integer linear programming approach. IEEE Trans Intell Transp Syst2022; 23(10): 17666–17676.
109.
CaoYSunYXieG, et al. A sound-based fault diagnosis method for railway point machines based on two-stage feature selection strategy and ensemble classifier. IEEE Trans Intell Transp Syst2022; 23(8): 12074–12083.
110.
CaoYWangZCLiuF, et al. Bio-inspired speed curve optimization and sliding mode tracking control for subway trains. IEEE Trans Vehicular Technol2019; 68(7): 6331–6342.
111.
CaoYJiYSunY, et al. The fault diagnosis of a switch machine based on deep random forest fusion. IEEE Intell Transp Syst Mag2023; 15(1): 437–452.
112.
DingFLiuYBaoB.Gradient-based and least-squares-based iterative estimation algorithms for multi-input multi-output systems. Proc IMechE, Part I: J Systems and Control Engineering2012; 226(1): 43–55.
113.
DingF.Combined state and least squares parameter estimation algorithms for dynamic systems. Appl Math Model2014; 38(1): 403–412.
114.
LiuL.Decomposition-based maximum likelihood gradient iterative algorithm for multivariate systems with colored noise. Int J Robust Nonlinear Control2024; 34(11): 7265–7284.
115.
LiuLLiFMaJ, et al. Auxiliary model maximum likelihood gradient-based iterative identification for feedback nonlinear systems. Optim Control Appl Methods2024; 45(5): 2346–2363.
116.
XuNDingFXuL.Convergence analysis of a synchronous gradient estimation scheme for time-varying parameter systems. J Comput Appl Math2023; 443: 115724.
117.
WangYLiuYChenJ, et al. Online identification of Hammerstein systems with B-spline networks. Int J Adapt Control Signal Process2024; 38(6): 2074–2092.
118.
XuLDingFZhangX, et al. Novel parameter estimation method for the systems with colored noises by using the filtering identification idea. Syst Control Lett2024; 186: 105774.
